tag:blogger.com,1999:blog-41262029649444010872024-02-20T09:26:49.682-08:00ppsekoshyavsamsofihttp://www.blogger.com/profile/12794342593186572847noreply@blogger.comBlogger14125tag:blogger.com,1999:blog-4126202964944401087.post-26944545961331434422021-02-24T20:10:00.001-08:002021-02-24T20:10:07.055-08:007)Sheikh Hasina the right choice as primary guest to the Republic Moment but trust the PMO to miss the obvious
<a href="https://indiarepublicday.com/"><b>India Republic Day</b></a> -- From a record 10 Chief Friends for the Republic Day Ornement in 2018 to non-e in 2021 is as a lot a reflection on Prime Minister Narendra Modis out of the box approach to foreign policy.
<br><br>
Coming from a record 10 Chief Friends for the Republic Day Ornement in 2018 to non-e in 2021 is as a lot a reflection on Prime Minister Narendra Modis out of the box approach to foreign policy as his blind spots although zeroing in on an astonishing foreign dignitary.
<br><br>
Sheikh Hasina Prime Minister of Bangladesh would have been the perfect Republic Day Chief Guest this season for umpteen reasons but it obviously didnt occur to Modi to single her out there for the honour. I shiver to even speculate if the visionary and statesman like Modi is blinded simply by her religion or sex or both to pass the girl up?
<br><br>
Instead of inviting United kingdom PM Boris Johnson who else ultimately chickened out Hasina should have been Modis programmed choice this year as it is the actual 50th anniversary of the birth of Bangladesh in 1971 and the 50th year of the establishment of diplomatic relations involving New Delhi and Dhaka. Indias role in the design of Bangladesh by bashing the Pakistani army is obviously well known. I have brought upward only to underline our buy-ins in Bangladesh. Moreover Hasina has been Indias steadfast in addition to unwavering ally in Southern Asia who has even crossed swords with Pakistan about Indias behalf. Her genuine commitment to New Delhi is a proven and identified fact. My friends in the Outer Affairs Ministry and the safety measures establishment tell me that that you of the biggest priorities of our own foreign policy is to make sure Hasina somehow remains the actual PM of Bangladesh which usually proves how invested we are in her.
<br><br>
Modi has a strange fixation for Initial World leaders. Last year this individual pulled out all stops to grab Donald Trump as the Primary Guest to show his home constituency that whether it is Barrack Obama or Trump zero US President can say no to Modi. But Trump was too frightened simply by pollution levels in Delhi or simply too bored with Modi to accept the invitation which usually resulted in an eleventh hour or so invite to the obnoxious in addition to repulsive Jair Bolsonaro associated with Brazil. In a sense Modi got the last laugh though. This individual lured Trump to Ahmedabad and Delhi in January 2020 with the promise connected with Gujarati votes in the US elections! This time Modi eyed the actual dishevelled Johnson who is furthermore also hard of reading. It was a done cope until it suddenly fell by so close to January 26 that finding another Primary Guest on the rebound seemed to be well and truly unattainable leaving Modi stranded.
<br><br>
Hasina has done so much for The indian subcontinent that Modi keeps having to pay Bangladesh compliment after supplement. He recently described the current phase of bilateral associations as the golden era connected with India-Bangladesh ties. Modi certainly doesnt talk out of the hat. He means just what he says. During a 90-minute long Virtual Summit with Hasina on December teen 2020 he said: Bangladesh is a major pillar connected with Indias Neighbourhood First coverage. From the very first day as PM strengthening and development of associations with Ban
gladesh has been a exclusive priority for me. Hasina immediately reciprocated by calling India Bangladeshs true friend.
<br><br>
There are far too many instances of these cordial exchanges. But I had been most touched by what Hasina said during a World Monetary Forum meeting in 2018 in Dalian. Our associations with India are natural. It cannot be measured by just a few billion dollars connected with trade. India and Bangladesh shed blood together to the creation of my region.
<br><br>
In Octobe
r 2020 India posted a new High Commissioner Bikram Doraiswami who else drove to Dhaka rather than flying there. While browsing Awami League headquarters about December 23 2020 Doraiswami according to a report in Ittefaq a leading Bangladeshi newspaper stated that if the Awami League isnt there The indian subcontinent will be friendless in Bangladesh. If thats indeed real doesnt Awami League soberano Hasina - who has a
sked Modi to Dhaka as being the Chief Guest on the occasion of the 50th anniversary connected with Bangladeshs independence in Walk 2021 - deserve to be the Chief Guest at our own Republic Day?samsofihttp://www.blogger.com/profile/12794342593186572847noreply@blogger.com0tag:blogger.com,1999:blog-4126202964944401087.post-91678066648421796082021-01-21T04:36:00.009-08:002021-01-21T04:36:31.917-08:00What Does Advertorial Placement On A Website Do?In this fast paced world where online business is the most common, it's important for you to use adverts wisely and in the correct places. There are several key principles you need to consider when placing advertorials - for example: don't place them too close to the product or service you're trying to sell. The product or service description should be enough to explain what it is. Don't overload your <a href="https://theadvertorial.com/">advertorial</a> with graphics - make sure you can keep your readers attention with less text. <br /><br />A recent case study showed how poorly an advertorial was placed may harm its success. A supermarket bought an advertorial for a local alpaca farm, placed it next to the products being sold. The ad was for the sale of 'Arai alpaca socks' and the article discussed the advantages of buying alpaca socks. Three months later, sales had fallen by 20% while the number of queries for the supermarket website had gone up. The ad was therefore ineffective in communicating the message that the company was selling quality alpaca socks. The supermarket's ad agency tried to place the advertorial in the opposite direction, putting it underneath the news article and below the fold. <br /><br />Other successful placements included in-store display, free on hire and newspaper advertising. For in-store displays, having an advertorial at the front of the store, working from the front to the back or inside the store and across the top of the page worked well. Free on hire work well if the offer is short term (a day or two) and if your target audience uses a smartphone, tablet or laptop to browse for news. Newspaper ads work well from the front to the bottom or inside the page - however, make sure your wording and the layout of the paper ads go with the look and feel of the website. <br /><br />Adverts that are placed in between articles on the same page of the news article tend to work well. Studies have shown that people read news articles alongside other content, so an advertorial under this situation may well entice customers to click through. Placing three or four adverts on a single page, in line with the other content, also works well. A further idea that works well in newspapers is having one or two advertorials on the first page of the news article, next to the other articles. <br /><br />For newspapers, placing adverts in the same area as the other content is generally successful. However, using text only adverts is not as effective. Studies have shown that people prefer to see information visually than reading it on a screen. If you use a combination of images and text, you can ensure that your advertorial will be read and, if necessary, follow the link to your website. <br /><br />If you are thinking about placing adverts in a sports magazine, you may be better off using an adverts page. This will enable you to place adverts at eye level, which increases the chances of the adverts being read. Similarly, adverts placed at the top of a news article will encourage readers to follow the link to your website. People are more likely to remember the adverts placed in the centre of a story and read the article in that way than they are to click on the adverts at the end of the story. <br /><br />However, adverts placed in the wrong place may actually discourage readers from clicking through to your website. If your adverts are too close to the edges of the content, the reader may be distracted and will more likely just cross the page to continue reading something else. Adverts that are too close to the bottom of the page will also cause distraction. Adverts that are placed in front of text will generally not be read, but may still cause irritation to the reader if they are situated too close to the edge of the text. Therefore, if your aim is for your advertverts to get the maximum exposure, it is better to place them where they are most effective. <br /><br />Advertorials work best when placed in places that will grab the reader's attention immediately. Adverts should be placed in high traffic areas of websites so that more people will see them. Adverts that are too close to the edges of the site will only result in people passing by your site without ever noticing your adverts. Adverts that are strategically placed in the middle of the page will capture the attention of many people because they will be searching for information regarding your product. It is a good idea to use several different adverts in order to find the ones that are most effective. <br /> samsofihttp://www.blogger.com/profile/12794342593186572847noreply@blogger.com0tag:blogger.com,1999:blog-4126202964944401087.post-87714810422169520342021-01-20T04:07:00.013-08:002021-01-20T04:07:30.372-08:00Angular momentum<img alt="" class="thumbimage" data-file-height="682" data-file-width="698" decoding="async" height="215" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/96/Ang_mom_2d.png/220px-Ang_mom_2d.png" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/96/Ang_mom_2d.png/330px-Ang_mom_2d.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/96/Ang_mom_2d.png/440px-Ang_mom_2d.png 2x" width="220"/><br/><br/><br/><p>In physics, <b>angular momentum</b> (rarely, <b>moment of momentum</b> or <b>rotational momentum</b>) is the rotational equivalent of linear momentum. It is an important quantity in physics because it is a conserved quantity—the total angular momentum of a closed system remains constant.
</p><p>In three dimensions, the angular momentum for a point particle is a pseudovector <span class="nowrap"><b>r</b> × <b>p</b></span>, the cross product of the particle's position vector <b>r</b> (relative to some origin) and its momentum vector; the latter is <span class="nowrap"><b>p</b> = <i>m</i><b>v</b></span> in Newtonian mechanics. This definition can be applied to each point in continua like solids or fluids, or physical fields. Unlike momentum, angular momentum does depend on where the origin is chosen, since the particle's position is measured from it.
</p><p>Just like for angular velocity, there are two special types of angular momentum: the spin angular momentum and orbital angular momentum. The spin angular momentum of an object is defined as the angular momentum about its centre of mass coordinate. The orbital angular momentum of an object about a chosen origin is defined as the angular momentum of the centre of mass about the origin. The total angular momentum of an object is the sum of the spin and orbital angular momenta. The orbital angular momentum vector of a point particle is always parallel and directly proportional to the orbital angular velocity vector <b>ω</b> of the particle, where the constant of proportionality depends on both the mass of the particle and its distance from origin. The spin angular momentum vector of a rigid body is proportional but not always parallel to the spin angular velocity vector <b>Ω</b>, making the constant of proportionality a second-rank tensor rather than a scalar.
</p><p>Angular momentum is an extensive quantity; i.e. the total angular momentum of any composite system is the sum of the angular momenta of its constituent parts. For a continuous rigid body, the total angular momentum is the volume integral of angular momentum density (i.e. angular momentum per unit volume in the limit as volume shrinks to zero) over the entire body.
</p><p>Torque can be defined as the rate of change of angular momentum, analogous to force. The net <i>external</i> torque on any system is always equal to the <i>total</i> torque on the system; in other words, the sum of all internal torques of any system is always 0 (this is the rotational analogue of Newton's Third Law). Therefore, for a <i>closed</i> system (where there is no net external torque), the <i>total</i> torque on the system must be 0, which means that the total angular momentum of the system is constant. The conservation of angular momentum helps explain many observed phenomena, for example the increase in rotational speed of a spinning figure skater as the skater's arms are contracted, the high rotational rates of neutron stars, the Coriolis effect, and the precession of gyroscopes. In general, conservation does limit the possible motion of a system, but does not uniquely determine what the exact motion is.
</p><p>In quantum mechanics, angular momentum (like other quantities) is expressed as an operator, and its one-dimensional projections have quantized eigenvalues. Angular momentum is subject to the Heisenberg uncertainty principle, implying that at any time, only one projection (also called "component") can be measured with definite precision; the other two then remain uncertain. Because of this, the notion of a quantum particle literally "spinning" about an axis does not exist. Quantum particles <i>do</i> possess a type of non-orbital angular momentum called "spin", but this angular momentum does not correspond to actual physical spinning motion.
</p>samsofihttp://www.blogger.com/profile/12794342593186572847noreply@blogger.com0tag:blogger.com,1999:blog-4126202964944401087.post-27597643845418495972021-01-20T04:07:00.011-08:002021-01-20T04:07:25.992-08:00Definition in classical mechanics<img alt="" class="thumbimage" data-file-height="682" data-file-width="698" decoding="async" height="215" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/96/Ang_mom_2d.png/220px-Ang_mom_2d.png" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/96/Ang_mom_2d.png/330px-Ang_mom_2d.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/96/Ang_mom_2d.png/440px-Ang_mom_2d.png 2x" width="220"/><br/><br/><br/><h3><span class="mw-headline" id="Orbital_angular_momentum_in_two_dimensions">Orbital angular momentum in two dimensions</span><span class="mw-editsection"><span class="mw-editsection-bracket"></span>edit<span class="mw-editsection-bracket"></span></span></h3><p>Angular momentum is a vector quantity (more precisely, a pseudovector) that represents the product of a body's rotational inertia and rotational velocity (in radians/sec) about a particular axis. However, if the particle's trajectory lies in a single plane, it is sufficient to discard the vector nature of angular momentum, and treat it as a scalar (more precisely, a pseudoscalar). Angular momentum can be considered a rotational analog of linear momentum. Thus, where linear momentum <span class="texhtml mvar" style="font-style:italic;">p</span> is proportional to mass <span class="texhtml mvar" style="font-style:italic;">m</span> and linear speed <span class="nowrap"><span class="texhtml mvar" style="font-style:italic;">v</span>,</span>
</p><dl><dd><span class="mwe-math-element" data-qid="Q41273"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle p=mv,}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>p</mi>
<mo>=</mo>
<mi>m</mi>
<mi>v</mi>
<mo>,</mo>
</mstyle>
</mrow>
{\displaystyle p=mv,}
</semantics>
</math></span><img alt="p=mv," aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb4b93eb594373b6fa496b126057996eefce3364" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:8.172ex; height:2.009ex;"/></span></dd></dl><p>angular momentum <span class="texhtml mvar" style="font-style:italic;">L</span> is proportional to moment of inertia <span class="texhtml mvar" style="font-style:italic;">I</span> and angular speed <span class="texhtml mvar" style="font-style:italic;">ω</span> measured in radians per second.
</p><dl><dd><span class="mwe-math-element" data-qid="Q161254"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle L=I\omega .}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>L</mi>
<mo>=</mo>
<mi>I</mi>
<mi>ω<!-- ω --></mi>
<mo>.</mo>
</mstyle>
</mrow>
{\displaystyle L=I\omega .}
</semantics>
</math></span><img alt="L=I\omega ." aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1ce1597dcef54047a562d9108835bcbd8358d32" style="vertical-align: -0.338ex; width:7.946ex; height:2.176ex;"/></span></dd></dl><p>Unlike mass, which depends only on amount of matter, moment of inertia is also dependent on the position of the axis of rotation and the shape of the matter. Unlike linear velocity, which does not depend upon the choice of origin, orbital angular velocity is always measured with respect to a fixed origin. Therefore, strictly speaking, <span class="texhtml mvar" style="font-style:italic;">L</span> should be referred to as the angular momentum <i>relative to that center</i>.
</p><p>Because <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle I=r^{2}m}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>I</mi>
<mo>=</mo>
<msup>
<mi>r</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mi>m</mi>
</mstyle>
</mrow>
{\displaystyle I=r^{2}m}
</semantics>
</math></span><img alt="{\displaystyle I=r^{2}m}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ce8ff961fab0cdfe7ac50e8bee6d6ef1ee62347" style="vertical-align: -0.338ex; width:8.413ex; height:2.676ex;"/></span> for a single particle and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle \omega ={\frac {v}{r}}}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ω<!-- ω --></mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>v</mi>
<mi>r</mi>
</mfrac>
</mrow>
</mstyle>
</mrow>
{\displaystyle \omega ={\frac {v}{r}}}
</semantics>
</math></span><img alt="\omega ={\frac {v}{r}}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12138afcc600a32b1607606c824d2064900871de" style="vertical-align: -1.838ex; width:6.508ex; height:4.676ex;"/></span> for circular motion, angular momentum can be expanded, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle L=r^{2}m\cdot {\frac {v}{r}},}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>L</mi>
<mo>=</mo>
<msup>
<mi>r</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mi>m</mi>
<mo>⋅<!-- ⋅ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>v</mi>
<mi>r</mi>
</mfrac>
</mrow>
<mo>,</mo>
</mstyle>
</mrow>
{\displaystyle L=r^{2}m\cdot {\frac {v}{r}},}
</semantics>
</math></span><img alt="{\displaystyle L=r^{2}m\cdot {\frac {v}{r}},}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/722877e31dc440b02e3df2b8c874eb991f25e4d0" style="vertical-align: -1.838ex; width:13.114ex; height:4.676ex;"/></span> and reduced to,
</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle L=rmv,}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>L</mi>
<mo>=</mo>
<mi>r</mi>
<mi>m</mi>
<mi>v</mi>
<mo>,</mo>
</mstyle>
</mrow>
{\displaystyle L=rmv,}
</semantics>
</math></span><img alt="{\displaystyle L=rmv,}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ce9be937fd731eccda69830a7cf2128deecb0ac" style="vertical-align: -0.671ex; width:9.545ex; height:2.509ex;"/></span></dd></dl><p>the product of the radius of rotation <span class="texhtml mvar" style="font-style:italic;">r</span> and the linear momentum of the particle <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle p=mv}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>p</mi>
<mo>=</mo>
<mi>m</mi>
<mi>v</mi>
</mstyle>
</mrow>
{\displaystyle p=mv}
</semantics>
</math></span><img alt="p = mv" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2acbe7154884d4dbe30b9a0b399e43cefd8654c7" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:7.525ex; height:2.009ex;"/></span>, where <span class="texhtml mvar" style="font-style:italic;">v</span> in this case is the equivalent linear (tangential) speed at the radius (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle =r\omega }" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo>=</mo>
<mi>r</mi>
<mi>ω<!-- ω --></mi>
</mstyle>
</mrow>
{\displaystyle =r\omega }
</semantics>
</math></span><img alt="{\displaystyle =r\omega }" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6e4dba86b9dc9a24f00a378577b4546dedc376e" style="vertical-align: -0.338ex; width:4.948ex; height:1.676ex;"/></span>).
</p><p>This simple analysis can also apply to non-circular motion if only the component of the motion which is perpendicular to the radius vector is considered. In that case,
</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle L=rmv_{\perp },}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>L</mi>
<mo>=</mo>
<mi>r</mi>
<mi>m</mi>
<msub>
<mi>v</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>⊥<!-- ⊥ --></mo>
</mrow>
</msub>
<mo>,</mo>
</mstyle>
</mrow>
{\displaystyle L=rmv_{\perp },}
</semantics>
</math></span><img alt="{\displaystyle L=rmv_{\perp },}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54cba40e6e82b32083735501d13f41e68cccf06d" style="vertical-align: -0.671ex; width:11.056ex; height:2.509ex;"/></span></dd></dl><p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle v_{\perp }=v\sin(\theta )}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>v</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>⊥<!-- ⊥ --></mo>
</mrow>
</msub>
<mo>=</mo>
<mi>v</mi>
<mi>sin</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mi>θ<!-- θ --></mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
{\displaystyle v_{\perp }=v\sin(\theta )}
</semantics>
</math></span><img alt="{\displaystyle v_{\perp }=v\sin(\theta )}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/163f5f644793a0f6eb62286bb838848317b0a8cb" style="vertical-align: -0.838ex; width:13.007ex; height:2.843ex;"/></span> is the perpendicular component of the motion. Expanding, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle L=rmv\sin(\theta ),}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>L</mi>
<mo>=</mo>
<mi>r</mi>
<mi>m</mi>
<mi>v</mi>
<mi>sin</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mi>θ<!-- θ --></mi>
<mo stretchy="false">)</mo>
<mo>,</mo>
</mstyle>
</mrow>
{\displaystyle L=rmv\sin(\theta ),}
</semantics>
</math></span><img alt="{\displaystyle L=rmv\sin(\theta ),}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f093a6fdb253e9d684af3f225239b92697cba364" style="vertical-align: -0.838ex; width:15.687ex; height:2.843ex;"/></span> rearranging, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle L=r\sin(\theta )mv,}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>L</mi>
<mo>=</mo>
<mi>r</mi>
<mi>sin</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mi>θ<!-- θ --></mi>
<mo stretchy="false">)</mo>
<mi>m</mi>
<mi>v</mi>
<mo>,</mo>
</mstyle>
</mrow>
{\displaystyle L=r\sin(\theta )mv,}
</semantics>
</math></span><img alt="{\displaystyle L=r\sin(\theta )mv,}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0e238548108c2e9e0c3ea696da8500c9885015c" style="vertical-align: -0.838ex; width:15.687ex; height:2.843ex;"/></span> and reducing, angular momentum can also be expressed,
</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle L=r_{\perp }mv,}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>L</mi>
<mo>=</mo>
<msub>
<mi>r</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>⊥<!-- ⊥ --></mo>
</mrow>
</msub>
<mi>m</mi>
<mi>v</mi>
<mo>,</mo>
</mstyle>
</mrow>
{\displaystyle L=r_{\perp }mv,}
</semantics>
</math></span><img alt="{\displaystyle L=r_{\perp }mv,}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb8528a226fa2915919bed41269ad720065d1352" style="vertical-align: -0.671ex; width:11.056ex; height:2.509ex;"/></span></dd></dl><p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle r_{\perp }=r\sin(\theta )}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>r</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>⊥<!-- ⊥ --></mo>
</mrow>
</msub>
<mo>=</mo>
<mi>r</mi>
<mi>sin</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mi>θ<!-- θ --></mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
{\displaystyle r_{\perp }=r\sin(\theta )}
</semantics>
</math></span><img alt="{\displaystyle r_{\perp }=r\sin(\theta )}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e429dae54c4064967213f2be1cf6170f814e664" style="vertical-align: -0.838ex; width:12.849ex; height:2.843ex;"/></span> is the length of the <i>moment arm</i>, a line dropped perpendicularly from the origin onto the path of the particle. It is this definition, <span class="texhtml">(length of moment arm)×(linear momentum)</span> to which the term <i>moment of momentum</i> refers.
</p><h3><span id="Scalar.E2.80.94angular_momentum_from_Lagrangian_mechanics"></span><span class="mw-headline" id="Scalar—angular_momentum_from_Lagrangian_mechanics">Scalar—angular momentum from Lagrangian mechanics</span><span class="mw-editsection"><span class="mw-editsection-bracket"></span>edit<span class="mw-editsection-bracket"></span></span></h3><p>Another approach is to define angular momentum as the conjugate momentum (also called <b>canonical momentum</b>) of the angular coordinate <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle \phi }" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ϕ<!-- ϕ --></mi>
</mstyle>
</mrow>
{\displaystyle \phi }
</semantics>
</math></span><img alt="\phi " aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;"/></span> expressed in the Lagrangian of the mechanical system. Consider a mechanical system with a mass <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle m}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>m</mi>
</mstyle>
</mrow>
{\displaystyle m}
</semantics>
</math></span><img alt="m" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;"/></span> constrained to move in a circle of radius <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle a}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>a</mi>
</mstyle>
</mrow>
{\displaystyle a}
</semantics>
</math></span><img alt="a" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;"/></span> in the absence of any external force field. The kinetic energy of the system is
</p><dl><dd><span class="mwe-math-element" data-qid="Q46276"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle T={\frac {1}{2}}ma^{2}\omega ^{2}={\frac {1}{2}}ma^{2}{\dot {\phi }}^{2}.}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>T</mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
<mi>m</mi>
<msup>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<msup>
<mi>ω<!-- ω --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
<mi>m</mi>
<msup>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>ϕ<!-- ϕ --></mi>
<mo>˙<!-- ˙ --></mo>
</mover>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>.</mo>
</mstyle>
</mrow>
{\displaystyle T={\frac {1}{2}}ma^{2}\omega ^{2}={\frac {1}{2}}ma^{2}{\dot {\phi }}^{2}.}
</semantics>
</math></span><img alt="{\displaystyle T={\frac {1}{2}}ma^{2}\omega ^{2}={\frac {1}{2}}ma^{2}{\dot {\phi }}^{2}.}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1966c3d1bc855abe77121a05dc8a9325040ad6de" style="vertical-align: -1.838ex; width:26.148ex; height:5.176ex;"/></span></dd></dl><p>And the potential energy is
</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle U=0.}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>U</mi>
<mo>=</mo>
<mn>0.</mn>
</mstyle>
</mrow>
{\displaystyle U=0.}
</semantics>
</math></span><img alt="{\displaystyle U=0.}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46118843389e8ab6108d1f62c9dcfe4afecefe16" style="vertical-align: -0.338ex; width:6.69ex; height:2.176ex;"/></span></dd></dl><p>Then the Lagrangian is
</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle {\mathcal {L}}\left(\phi ,{\dot {\phi }}\right)=T-U={\frac {1}{2}}ma^{2}{\dot {\phi }}^{2}.}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">L</mi>
</mrow>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>ϕ<!-- ϕ --></mi>
<mo>,</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>ϕ<!-- ϕ --></mi>
<mo>˙<!-- ˙ --></mo>
</mover>
</mrow>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mi>T</mi>
<mo>−<!-- − --></mo>
<mi>U</mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
<mi>m</mi>
<msup>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>ϕ<!-- ϕ --></mi>
<mo>˙<!-- ˙ --></mo>
</mover>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>.</mo>
</mstyle>
</mrow>
{\displaystyle {\mathcal {L}}\left(\phi ,{\dot {\phi }}\right)=T-U={\frac {1}{2}}ma^{2}{\dot {\phi }}^{2}.}
</semantics>
</math></span><img alt="{\displaystyle {\mathcal {L}}\left(\phi ,{\dot {\phi }}\right)=T-U={\frac {1}{2}}ma^{2}{\dot {\phi }}^{2}.}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebeb14b16b98f192a121c3c27490679c5e3bfbb4" style="vertical-align: -1.838ex; width:30.601ex; height:5.176ex;"/></span></dd></dl><p>The <i>generalized momentum</i> "canonically conjugate to" the coordinate <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle \phi }" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ϕ<!-- ϕ --></mi>
</mstyle>
</mrow>
{\displaystyle \phi }
</semantics>
</math></span><img alt="\phi " aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;"/></span> is defined by
</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle p_{\phi }={\frac {\partial {\mathcal {L}}}{\partial {\dot {\phi }}}}=ma^{2}{\dot {\phi }}=I\omega =L.}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>ϕ<!-- ϕ --></mi>
</mrow>
</msub>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">L</mi>
</mrow>
</mrow>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>ϕ<!-- ϕ --></mi>
<mo>˙<!-- ˙ --></mo>
</mover>
</mrow>
</mrow>
</mrow>
</mfrac>
</mrow>
<mo>=</mo>
<mi>m</mi>
<msup>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>ϕ<!-- ϕ --></mi>
<mo>˙<!-- ˙ --></mo>
</mover>
</mrow>
</mrow>
<mo>=</mo>
<mi>I</mi>
<mi>ω<!-- ω --></mi>
<mo>=</mo>
<mi>L</mi>
<mo>.</mo>
</mstyle>
</mrow>
{\displaystyle p_{\phi }={\frac {\partial {\mathcal {L}}}{\partial {\dot {\phi }}}}=ma^{2}{\dot {\phi }}=I\omega =L.}
</semantics>
</math></span><img alt="{\displaystyle p_{\phi }={\frac {\partial {\mathcal {L}}}{\partial {\dot {\phi }}}}=ma^{2}{\dot {\phi }}=I\omega =L.}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3ba5987770cca1c980cd306463d3f2271ac9761" style="vertical-align: -3.005ex; margin-left: -0.089ex; width:29.262ex; height:6.509ex;"/></span></dd></dl><h3><span class="mw-headline" id="Orbital_angular_momentum_in_three_dimensions">Orbital angular momentum in three dimensions</span><span class="mw-editsection"><span class="mw-editsection-bracket"></span>edit<span class="mw-editsection-bracket"></span></span></h3><p>To completely define orbital angular momentum in three dimensions, it is required to know the rate at which the position vector sweeps out angle, the direction perpendicular to the instantaneous plane of angular displacement, and the mass involved, as well as how this mass is distributed in space. By retaining this vector nature of angular momentum, the general nature of the equations is also retained, and can describe any sort of three-dimensional motion about the center of rotation – circular, linear, or otherwise. In vector notation, the orbital angular momentum of a point particle in motion about the origin can be expressed as:
</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle \mathbf {L} =I{\boldsymbol {\omega }},}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">L</mi>
</mrow>
<mo>=</mo>
<mi>I</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold-italic">ω<!-- ω --></mi>
</mrow>
<mo>,</mo>
</mstyle>
</mrow>
{\displaystyle \mathbf {L} =I{\boldsymbol {\omega }},}
</semantics>
</math></span><img alt="\mathbf {L} =I{\boldsymbol {\omega }}," aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afb18c91ec4284e8e4142e3b7fea1b3cb54baf40" style="vertical-align: -0.671ex; width:8.194ex; height:2.509ex;"/></span></dd></dl><p>where
</p><dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle I=r^{2}m}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>I</mi>
<mo>=</mo>
<msup>
<mi>r</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mi>m</mi>
</mstyle>
</mrow>
{\displaystyle I=r^{2}m}
</semantics>
</math></span><img alt="{\displaystyle I=r^{2}m}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ce8ff961fab0cdfe7ac50e8bee6d6ef1ee62347" style="vertical-align: -0.338ex; width:8.413ex; height:2.676ex;"/></span> is the moment of inertia for a point mass,</dd>
<dd><span class="mwe-math-element" data-qid="Q161635"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle {\boldsymbol {\omega }}={\frac {\mathbf {r} \times \mathbf {v} }{r^{2}}}}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold-italic">ω<!-- ω --></mi>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">r</mi>
</mrow>
<mo>×<!-- × --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">v</mi>
</mrow>
</mrow>
<msup>
<mi>r</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mfrac>
</mrow>
</mstyle>
</mrow>
{\displaystyle {\boldsymbol {\omega }}={\frac {\mathbf {r} \times \mathbf {v} }{r^{2}}}}
</semantics>
</math></span><img alt="{\boldsymbol {\omega }}={\frac {\mathbf {r} \times \mathbf {v} }{r^{2}}}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d106050e9a58068eb301336636fb3aae9ac540f6" style="vertical-align: -2.171ex; width:10.957ex; height:5.176ex;"/></span> is the orbital angular velocity in radians/sec (units 1/sec) of the particle about the origin,</dd>
<dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle \mathbf {r} }" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">r</mi>
</mrow>
</mstyle>
</mrow>
{\displaystyle \mathbf {r} }
</semantics>
</math></span><img alt="\mathbf {r} " aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eca0f46511c4c986c48b254073732c0bd98ae0c1" style="vertical-align: -0.338ex; width:1.102ex; height:1.676ex;"/></span> is the position vector of the particle relative to the origin, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle r=\left\vert \mathbf {r} \right\vert }" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>r</mi>
<mo>=</mo>
<mrow>
<mo>|</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">r</mi>
</mrow>
<mo>|</mo>
</mrow>
</mstyle>
</mrow>
{\displaystyle r=\left\vert \mathbf {r} \right\vert }
</semantics>
</math></span><img alt="r=\left\vert \mathbf {r} \right\vert " aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/188bf31840a7fed7f6b36f9c107664c5881a7969" style="vertical-align: -0.838ex; width:6.543ex; height:2.843ex;"/></span>,</dd>
<dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle \mathbf {v} }" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">v</mi>
</mrow>
</mstyle>
</mrow>
{\displaystyle \mathbf {v} }
</semantics>
</math></span><img alt="\mathbf {v} " aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35c1866e359fbfd2e0f606c725ba5cc37a5195d6" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;"/></span> is the linear velocity of the particle relative to the origin, and</dd>
<dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle m}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>m</mi>
</mstyle>
</mrow>
{\displaystyle m}
</semantics>
</math></span><img alt="m" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;"/></span> is the mass of the particle.</dd></dl></dd></dl><p>This can be expanded, reduced, and by the rules of vector algebra, rearranged:
</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle {\begin{aligned}\mathbf {L} &=\left(r^{2}m\right)\left({\frac {\mathbf {r} \times \mathbf {v} }{r^{2}}}\right)\\&=m\left(\mathbf {r} \times \mathbf {v} \right)\\&=\mathbf {r} \times m\mathbf {v} \\&=\mathbf {r} \times \mathbf {p} ,\end{aligned}}}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mtable columnalign="right left right left right left right left right left right left" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true" rowspacing="3pt">
<mtr>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">L</mi>
</mrow>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<msup>
<mi>r</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mi>m</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">r</mi>
</mrow>
<mo>×<!-- × --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">v</mi>
</mrow>
</mrow>
<msup>
<mi>r</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd></mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mi>m</mi>
<mrow>
<mo>(</mo>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">r</mi>
</mrow>
<mo>×<!-- × --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">v</mi>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd></mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">r</mi>
</mrow>
<mo>×<!-- × --></mo>
<mi>m</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">v</mi>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd></mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">r</mi>
</mrow>
<mo>×<!-- × --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">p</mi>
</mrow>
<mo>,</mo>
</mtd>
</mtr>
</mtable>
</mrow>
</mstyle>
</mrow>
{\displaystyle {\begin{aligned}\mathbf {L} &=\left(r^{2}m\right)\left({\frac {\mathbf {r} \times \mathbf {v} }{r^{2}}}\right)\\&=m\left(\mathbf {r} \times \mathbf {v} \right)\\&=\mathbf {r} \times m\mathbf {v} \\&=\mathbf {r} \times \mathbf {p} ,\end{aligned}}}
</semantics>
</math></span><img alt="{\displaystyle {\begin{aligned}\mathbf {L} &=\left(r^{2}m\right)\left({\frac {\mathbf {r} \times \mathbf {v} }{r^{2}}}\right)\\&=m\left(\mathbf {r} \times \mathbf {v} \right)\\&=\mathbf {r} \times m\mathbf {v} \\&=\mathbf {r} \times \mathbf {p} ,\end{aligned}}}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a06879332a77f41eab2433dce133a996b34e0716" style="vertical-align: -7.005ex; width:21.729ex; height:15.176ex;"/></span></dd></dl><p>which is the cross product of the position vector <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle \mathbf {r} }" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">r</mi>
</mrow>
</mstyle>
</mrow>
{\displaystyle \mathbf {r} }
</semantics>
</math></span><img alt="\mathbf {r} " aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eca0f46511c4c986c48b254073732c0bd98ae0c1" style="vertical-align: -0.338ex; width:1.102ex; height:1.676ex;"/></span> and the linear momentum <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle \mathbf {p} =m\mathbf {v} }" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">p</mi>
</mrow>
<mo>=</mo>
<mi>m</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">v</mi>
</mrow>
</mstyle>
</mrow>
{\displaystyle \mathbf {p} =m\mathbf {v} }
</semantics>
</math></span><img alt="{\mathbf {p}}=m{\mathbf {v}}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a271a96e7b925fd39686375167c76d406e87c813" style="vertical-align: -0.671ex; width:8.035ex; height:2.009ex;"/></span> of the particle. By the definition of the cross product, the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle \mathbf {L} }" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">L</mi>
</mrow>
</mstyle>
</mrow>
{\displaystyle \mathbf {L} }
</semantics>
</math></span><img alt="\mathbf {L} " aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f5f750865376a1a4ae2b15a00b4ff9c75a66630" style="vertical-align: -0.338ex; width:1.608ex; height:2.176ex;"/></span> vector is perpendicular to both <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle \mathbf {r} }" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">r</mi>
</mrow>
</mstyle>
</mrow>
{\displaystyle \mathbf {r} }
</semantics>
</math></span><img alt="\mathbf {r} " aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eca0f46511c4c986c48b254073732c0bd98ae0c1" style="vertical-align: -0.338ex; width:1.102ex; height:1.676ex;"/></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle \mathbf {p} }" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">p</mi>
</mrow>
</mstyle>
</mrow>
{\displaystyle \mathbf {p} }
</semantics>
</math></span><img alt="\mathbf {p} " aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd73e3862cb92b016721b8c492eadb4e8a577527" style="vertical-align: -0.671ex; width:1.485ex; height:2.009ex;"/></span>. It is directed perpendicular to the plane of angular displacement, as indicated by the right-hand rule – so that the angular velocity is seen as counter-clockwise from the head of the vector. Conversely, the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle \mathbf {L} }" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">L</mi>
</mrow>
</mstyle>
</mrow>
{\displaystyle \mathbf {L} }
</semantics>
</math></span><img alt="\mathbf {L} " aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f5f750865376a1a4ae2b15a00b4ff9c75a66630" style="vertical-align: -0.338ex; width:1.608ex; height:2.176ex;"/></span> vector defines the plane in which <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle \mathbf {r} }" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">r</mi>
</mrow>
</mstyle>
</mrow>
{\displaystyle \mathbf {r} }
</semantics>
</math></span><img alt="\mathbf {r} " aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eca0f46511c4c986c48b254073732c0bd98ae0c1" style="vertical-align: -0.338ex; width:1.102ex; height:1.676ex;"/></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle \mathbf {p} }" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">p</mi>
</mrow>
</mstyle>
</mrow>
{\displaystyle \mathbf {p} }
</semantics>
</math></span><img alt="\mathbf {p} " aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd73e3862cb92b016721b8c492eadb4e8a577527" style="vertical-align: -0.671ex; width:1.485ex; height:2.009ex;"/></span> lie.
</p><p>By defining a unit vector <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle \mathbf {\hat {u}} }" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi mathvariant="bold">u</mi>
<mo mathvariant="bold" stretchy="false">^<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
</mstyle>
</mrow>
{\displaystyle \mathbf {\hat {u}} }
</semantics>
</math></span><img alt="\mathbf {\hat {u}} " aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1adccb18b3b18c193af9f9ca2b0c0c500103e3ea" style="vertical-align: -0.338ex; width:1.485ex; height:2.343ex;"/></span> perpendicular to the plane of angular displacement, a scalar angular speed <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle \omega }" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ω<!-- ω --></mi>
</mstyle>
</mrow>
{\displaystyle \omega }
</semantics>
</math></span><img alt="\omega " aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;"/></span> results, where
</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle \omega \mathbf {\hat {u}} ={\boldsymbol {\omega }},}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ω<!-- ω --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi mathvariant="bold">u</mi>
<mo mathvariant="bold" stretchy="false">^<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold-italic">ω<!-- ω --></mi>
</mrow>
<mo>,</mo>
</mstyle>
</mrow>
{\displaystyle \omega \mathbf {\hat {u}} ={\boldsymbol {\omega }},}
</semantics>
</math></span><img alt="{\displaystyle \omega \mathbf {\hat {u}} ={\boldsymbol {\omega }},}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc410d6366c23541fe0100e618a54d8f293a6b55" style="vertical-align: -0.671ex; width:8.345ex; height:2.676ex;"/></span> and</dd>
<dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle \omega ={\frac {v_{\perp }}{r}},}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ω<!-- ω --></mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<msub>
<mi>v</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>⊥<!-- ⊥ --></mo>
</mrow>
</msub>
<mi>r</mi>
</mfrac>
</mrow>
<mo>,</mo>
</mstyle>
</mrow>
{\displaystyle \omega ={\frac {v_{\perp }}{r}},}
</semantics>
</math></span><img alt="{\displaystyle \omega ={\frac {v_{\perp }}{r}},}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95c29a5398be29c77b66df211e844987301487fe" style="vertical-align: -1.838ex; width:8.666ex; height:4.676ex;"/></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle v_{\perp }}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>v</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>⊥<!-- ⊥ --></mo>
</mrow>
</msub>
</mstyle>
</mrow>
{\displaystyle v_{\perp }}
</semantics>
</math></span><img alt="v_\perp" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f80a8cf80254aa3ef2640555e94986487d5cba0b" style="vertical-align: -0.671ex; width:2.638ex; height:2.009ex;"/></span> is the perpendicular component of the motion, as above.</dd></dl><p>The two-dimensional scalar equations of the previous section can thus be given direction:
</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle {\begin{aligned}\mathbf {L} &=I{\boldsymbol {\omega }}\\&=I\omega \mathbf {\hat {u}} \\&=\left(r^{2}m\right)\omega \mathbf {\hat {u}} \\&=rmv_{\perp }\mathbf {\hat {u}} \\&=r_{\perp }mv\mathbf {\hat {u}} ,\end{aligned}}}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mtable columnalign="right left right left right left right left right left right left" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true" rowspacing="3pt">
<mtr>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">L</mi>
</mrow>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mi>I</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold-italic">ω<!-- ω --></mi>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd></mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mi>I</mi>
<mi>ω<!-- ω --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi mathvariant="bold">u</mi>
<mo mathvariant="bold" stretchy="false">^<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd></mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<msup>
<mi>r</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mi>m</mi>
</mrow>
<mo>)</mo>
</mrow>
<mi>ω<!-- ω --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi mathvariant="bold">u</mi>
<mo mathvariant="bold" stretchy="false">^<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd></mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mi>r</mi>
<mi>m</mi>
<msub>
<mi>v</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>⊥<!-- ⊥ --></mo>
</mrow>
</msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi mathvariant="bold">u</mi>
<mo mathvariant="bold" stretchy="false">^<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd></mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<msub>
<mi>r</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>⊥<!-- ⊥ --></mo>
</mrow>
</msub>
<mi>m</mi>
<mi>v</mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi mathvariant="bold">u</mi>
<mo mathvariant="bold" stretchy="false">^<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
<mo>,</mo>
</mtd>
</mtr>
</mtable>
</mrow>
</mstyle>
</mrow>
{\displaystyle {\begin{aligned}\mathbf {L} &=I{\boldsymbol {\omega }}\\&=I\omega \mathbf {\hat {u}} \\&=\left(r^{2}m\right)\omega \mathbf {\hat {u}} \\&=rmv_{\perp }\mathbf {\hat {u}} \\&=r_{\perp }mv\mathbf {\hat {u}} ,\end{aligned}}}
</semantics>
</math></span><img alt="{\displaystyle {\begin{aligned}\mathbf {L} &=I{\boldsymbol {\omega }}\\&=I\omega \mathbf {\hat {u}} \\&=\left(r^{2}m\right)\omega \mathbf {\hat {u}} \\&=rmv_{\perp }\mathbf {\hat {u}} \\&=r_{\perp }mv\mathbf {\hat {u}} ,\end{aligned}}}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/138200e265424b4778f846322c7d77ae28571d5e" style="vertical-align: -7.171ex; width:15.05ex; height:15.509ex;"/></span></dd></dl><p>and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle \mathbf {L} =rmv\mathbf {\hat {u}} }" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">L</mi>
</mrow>
<mo>=</mo>
<mi>r</mi>
<mi>m</mi>
<mi>v</mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi mathvariant="bold">u</mi>
<mo mathvariant="bold" stretchy="false">^<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
</mstyle>
</mrow>
{\displaystyle \mathbf {L} =rmv\mathbf {\hat {u}} }
</semantics>
</math></span><img alt="{\displaystyle \mathbf {L} =rmv\mathbf {\hat {u}} }" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a761fad402cde98517baffffd37adae166f765d" style="vertical-align: -0.338ex; width:10.409ex; height:2.343ex;"/></span> for circular motion, where all of the motion is perpendicular to the radius <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle r}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>r</mi>
</mstyle>
</mrow>
{\displaystyle r}
</semantics>
</math></span><img alt="r" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;"/></span>.
</p><p>In the spherical coordinate system the angular momentum vector expresses as
</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle \mathbf {L} =m\mathbf {r} \times \mathbf {v} =mr^{2}\left({\dot {\theta }}\,{\hat {\boldsymbol {\varphi }}}-{\dot {\varphi }}\sin \theta \,\mathbf {\hat {\boldsymbol {\theta }}} \right).}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">L</mi>
</mrow>
<mo>=</mo>
<mi>m</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">r</mi>
</mrow>
<mo>×<!-- × --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">v</mi>
</mrow>
<mo>=</mo>
<mi>m</mi>
<msup>
<mi>r</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mrow>
<mo>(</mo>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>θ<!-- θ --></mi>
<mo>˙<!-- ˙ --></mo>
</mover>
</mrow>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi mathvariant="bold-italic">φ<!-- φ --></mi>
<mo stretchy="false">^<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>φ<!-- φ --></mi>
<mo>˙<!-- ˙ --></mo>
</mover>
</mrow>
</mrow>
<mi>sin</mi>
<mo><!-- --></mo>
<mi>θ<!-- θ --></mi>
<mspace width="thinmathspace"></mspace>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi mathvariant="bold-italic">θ<!-- θ --></mi>
<mo mathvariant="bold" stretchy="false">^<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>.</mo>
</mstyle>
</mrow>
{\displaystyle \mathbf {L} =m\mathbf {r} \times \mathbf {v} =mr^{2}\left({\dot {\theta }}\,{\hat {\boldsymbol {\varphi }}}-{\dot {\varphi }}\sin \theta \,\mathbf {\hat {\boldsymbol {\theta }}} \right).}
</semantics>
</math></span><img alt="{\displaystyle \mathbf {L} =m\mathbf {r} \times \mathbf {v} =mr^{2}\left({\dot {\theta }}\,{\hat {\boldsymbol {\varphi }}}-{\dot {\varphi }}\sin \theta \,\mathbf {\hat {\boldsymbol {\theta }}} \right).}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95551fade90d8e24bb3588cb75aa703a8d8e843e" style="vertical-align: -1.838ex; width:38.06ex; height:4.843ex;"/></span></dd></dl>samsofihttp://www.blogger.com/profile/12794342593186572847noreply@blogger.com0tag:blogger.com,1999:blog-4126202964944401087.post-46597732347511175982021-01-20T04:07:00.009-08:002021-01-20T04:07:20.721-08:00Analogy to linear momentum<img alt="" class="thumbimage" data-file-height="480" data-file-width="640" decoding="async" height="165" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ad/Moment_of_inertia_examples.gif/220px-Moment_of_inertia_examples.gif" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ad/Moment_of_inertia_examples.gif/330px-Moment_of_inertia_examples.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/ad/Moment_of_inertia_examples.gif/440px-Moment_of_inertia_examples.gif 2x" width="220"/><br/><br/><br/><p>Angular momentum can be described as the rotational analog of linear momentum. Like linear momentum it involves elements of mass and displacement. Unlike linear momentum it also involves elements of position and shape.
</p><p>Many problems in physics involve matter in motion about some certain point in space, be it in actual rotation about it, or simply moving past it, where it is desired to know what effect the moving matter has on the point—can it exert energy upon it or perform work about it? Energy, the ability to do work, can be stored in matter by setting it in motion—a combination of its inertia and its displacement. Inertia is measured by its mass, and displacement by its velocity. Their product,
</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle {\begin{aligned}({\text{amount of inertia}})\times ({\text{amount of displacement}})&={\text{amount of (inertia⋅displacement)}}\\{\text{mass}}\times {\text{velocity}}&={\text{momentum}}\\m\times v&=p\\\end{aligned}}}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mtable columnalign="right left right left right left right left right left right left" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true" rowspacing="3pt">
<mtr>
<mtd>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mtext>amount of inertia</mtext>
</mrow>
<mo stretchy="false">)</mo>
<mo>×<!-- × --></mo>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mtext>amount of displacement</mtext>
</mrow>
<mo stretchy="false">)</mo>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mtext>amount of (inertia⋅displacement)</mtext>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mtext>mass</mtext>
</mrow>
<mo>×<!-- × --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mtext>velocity</mtext>
</mrow>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mtext>momentum</mtext>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>m</mi>
<mo>×<!-- × --></mo>
<mi>v</mi>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mi>p</mi>
</mtd>
</mtr>
</mtable>
</mrow>
</mstyle>
</mrow>
{\displaystyle {\begin{aligned}({\text{amount of inertia}})\times ({\text{amount of displacement}})&={\text{amount of (inertia⋅displacement)}}\\{\text{mass}}\times {\text{velocity}}&={\text{momentum}}\\m\times v&=p\\\end{aligned}}}
</semantics>
</math></span><img alt="{\displaystyle {\begin{aligned}({\text{amount of inertia}})\times ({\text{amount of displacement}})&={\text{amount of (inertia⋅displacement)}}\\{\text{mass}}\times {\text{velocity}}&={\text{momentum}}\\m\times v&=p\\\end{aligned}}}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b0cb2452f95a0f974572985f205329a1d55bc71" style="vertical-align: -3.838ex; width:84.693ex; height:8.843ex;"/></span></dd></dl><p>is the matter's momentum. Referring this momentum to a central point introduces a complication: the momentum is not applied to the point directly. For instance, a particle of matter at the outer edge of a wheel is, in effect, at the end of a lever of the same length as the wheel's radius, its momentum turning the lever about the center point. This imaginary lever is known as the <i>moment arm</i>. It has the effect of multiplying the momentum's effort in proportion to its length, an effect known as a <i>moment</i>. Hence, the particle's momentum referred to a particular point,
</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle {\begin{aligned}({\text{moment arm}})\times ({\text{amount of inertia}})\times ({\text{amount of displacement}})&={\text{moment of (inertia⋅displacement)}}\\{\text{length}}\times {\text{mass}}\times {\text{velocity}}&={\text{moment of momentum}}\\r\times m\times v&=L\\\end{aligned}}}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mtable columnalign="right left right left right left right left right left right left" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true" rowspacing="3pt">
<mtr>
<mtd>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mtext>moment arm</mtext>
</mrow>
<mo stretchy="false">)</mo>
<mo>×<!-- × --></mo>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mtext>amount of inertia</mtext>
</mrow>
<mo stretchy="false">)</mo>
<mo>×<!-- × --></mo>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mtext>amount of displacement</mtext>
</mrow>
<mo stretchy="false">)</mo>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mtext>moment of (inertia⋅displacement)</mtext>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mtext>length</mtext>
</mrow>
<mo>×<!-- × --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mtext>mass</mtext>
</mrow>
<mo>×<!-- × --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mtext>velocity</mtext>
</mrow>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mtext>moment of momentum</mtext>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>r</mi>
<mo>×<!-- × --></mo>
<mi>m</mi>
<mo>×<!-- × --></mo>
<mi>v</mi>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mi>L</mi>
</mtd>
</mtr>
</mtable>
</mrow>
</mstyle>
</mrow>
{\displaystyle {\begin{aligned}({\text{moment arm}})\times ({\text{amount of inertia}})\times ({\text{amount of displacement}})&={\text{moment of (inertia⋅displacement)}}\\{\text{length}}\times {\text{mass}}\times {\text{velocity}}&={\text{moment of momentum}}\\r\times m\times v&=L\\\end{aligned}}}
</semantics>
</math></span><img alt="{\displaystyle {\begin{aligned}({\text{moment arm}})\times ({\text{amount of inertia}})\times ({\text{amount of displacement}})&={\text{moment of (inertia⋅displacement)}}\\{\text{length}}\times {\text{mass}}\times {\text{velocity}}&={\text{moment of momentum}}\\r\times m\times v&=L\\\end{aligned}}}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4002dc08fbcabcd33066cf654e7fd49e1adc402e" style="vertical-align: -4.005ex; width:102.711ex; height:9.009ex;"/></span></dd></dl><p>is the <i>angular momentum</i>, sometimes called, as here, the <i>moment of momentum</i> of the particle versus that particular center point. The equation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle L=rmv}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>L</mi>
<mo>=</mo>
<mi>r</mi>
<mi>m</mi>
<mi>v</mi>
</mstyle>
</mrow>
{\displaystyle L=rmv}
</semantics>
</math></span><img alt="L=rmv" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cae824c40f5d6282d0a4d286c1bdb7a7382c925e" style="vertical-align: -0.338ex; width:8.898ex; height:2.176ex;"/></span> combines a moment (a mass <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle m}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>m</mi>
</mstyle>
</mrow>
{\displaystyle m}
</semantics>
</math></span><img alt="m" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;"/></span> turning moment arm <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle r}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>r</mi>
</mstyle>
</mrow>
{\displaystyle r}
</semantics>
</math></span><img alt="r" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;"/></span>) with a linear (straight-line equivalent) speed <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle v}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>v</mi>
</mstyle>
</mrow>
{\displaystyle v}
</semantics>
</math></span><img alt="v" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;"/></span>. Linear speed referred to the central point is simply the product of the distance <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle r}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>r</mi>
</mstyle>
</mrow>
{\displaystyle r}
</semantics>
</math></span><img alt="r" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;"/></span> and the angular speed <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle \omega }" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ω<!-- ω --></mi>
</mstyle>
</mrow>
{\displaystyle \omega }
</semantics>
</math></span><img alt="\omega " aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;"/></span> versus the point: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle v=r\omega ,}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>v</mi>
<mo>=</mo>
<mi>r</mi>
<mi>ω<!-- ω --></mi>
<mo>,</mo>
</mstyle>
</mrow>
{\displaystyle v=r\omega ,}
</semantics>
</math></span><img alt="v=r\omega ," aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/574eabd58d8470d1f1eded75c6b545fb35f48d5c" style="vertical-align: -0.671ex; width:7.367ex; height:2.009ex;"/></span> another moment. Hence, angular momentum contains a double moment: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle L=rmr\omega .}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>L</mi>
<mo>=</mo>
<mi>r</mi>
<mi>m</mi>
<mi>r</mi>
<mi>ω<!-- ω --></mi>
<mo>.</mo>
</mstyle>
</mrow>
{\displaystyle L=rmr\omega .}
</semantics>
</math></span><img alt="L=rmr\omega ." aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e34e11145331181be5f931f586322cd5b0a81aee" style="vertical-align: -0.338ex; width:10.912ex; height:2.176ex;"/></span> Simplifying slightly, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle L=r^{2}m\omega ,}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>L</mi>
<mo>=</mo>
<msup>
<mi>r</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mi>m</mi>
<mi>ω<!-- ω --></mi>
<mo>,</mo>
</mstyle>
</mrow>
{\displaystyle L=r^{2}m\omega ,}
</semantics>
</math></span><img alt="L=r^{2}m\omega ," aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d901cacb8e9e59ef9ac4d17913b6d7922c47808f" style="vertical-align: -0.671ex; width:10.917ex; height:3.009ex;"/></span> the quantity <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle r^{2}m}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>r</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mi>m</mi>
</mstyle>
</mrow>
{\displaystyle r^{2}m}
</semantics>
</math></span><img alt="r^{2}m" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1bbbb15eaf46c7e295eefc1ceb32d90161f7bac4" style="vertical-align: -0.338ex; width:4.143ex; height:2.676ex;"/></span> is the particle's moment of inertia, sometimes called the second moment of mass. It is a measure of rotational inertia.
</p><p>Because moment of inertia is a crucial part of the spin angular momentum, the latter necessarily includes all of the complications of the former, which is calculated by multiplying elementary bits of the mass by the squares of their distances from the center of rotation. Therefore, the total moment of inertia, and the angular momentum, is a complex function of the configuration of the matter about the center of rotation and the orientation of the rotation for the various bits.
</p><p>For a rigid body, for instance a wheel or an asteroid, the orientation of rotation is simply the position of the rotation axis versus the matter of the body. It may or may not pass through the center of mass, or it may lie completely outside of the body. For the same body, angular momentum may take a different value for every possible axis about which rotation may take place. It reaches a minimum when the axis passes through the center of mass.
</p><p>For a collection of objects revolving about a center, for instance all of the bodies of the Solar System, the orientations may be somewhat organized, as is the Solar System, with most of the bodies' axes lying close to the system's axis. Their orientations may also be completely random.
</p><p>In brief, the more mass and the farther it is from the center of rotation (the longer the moment arm), the greater the moment of inertia, and therefore the greater the angular momentum for a given angular velocity. In many cases the moment of inertia, and hence the angular momentum, can be simplified by,
</p><dl><dd><span class="mwe-math-element" data-qid="Q165618"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle I=k^{2}m,}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>I</mi>
<mo>=</mo>
<msup>
<mi>k</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mi>m</mi>
<mo>,</mo>
</mstyle>
</mrow>
{\displaystyle I=k^{2}m,}
</semantics>
</math></span><img alt="I=k^{2}m," aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9262e5be6a71693fea700ba53795be70fde6291c" style="vertical-align: -0.671ex; width:9.223ex; height:3.009ex;"/></span></dd>
<dd>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle k}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>k</mi>
</mstyle>
</mrow>
{\displaystyle k}
</semantics>
</math></span><img alt="k" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;"/></span> is the radius of gyration, the distance from the axis at which the entire mass <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle m}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>m</mi>
</mstyle>
</mrow>
{\displaystyle m}
</semantics>
</math></span><img alt="m" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;"/></span> may be considered as concentrated.</dd></dl><p>Similarly, for a point mass <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle m}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>m</mi>
</mstyle>
</mrow>
{\displaystyle m}
</semantics>
</math></span><img alt="m" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;"/></span> the moment of inertia is defined as,
</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle I=r^{2}m}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>I</mi>
<mo>=</mo>
<msup>
<mi>r</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mi>m</mi>
</mstyle>
</mrow>
{\displaystyle I=r^{2}m}
</semantics>
</math></span><img alt="I=r^{2}m" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ce8ff961fab0cdfe7ac50e8bee6d6ef1ee62347" style="vertical-align: -0.338ex; width:8.413ex; height:2.676ex;"/></span></dd>
<dd>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle r}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>r</mi>
</mstyle>
</mrow>
{\displaystyle r}
</semantics>
</math></span><img alt="r" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;"/></span> is the radius of the point mass from the center of rotation,</dd></dl><p>and for any collection of particles <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle m_{i}}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>m</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mstyle>
</mrow>
{\displaystyle m_{i}}
</semantics>
</math></span><img alt="m_{i}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95ec8e804f69706d3f5ad235f4f983220c8df7c2" style="vertical-align: -0.671ex; width:2.84ex; height:2.009ex;"/></span> as the sum,
</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle \sum _{i}I_{i}=\sum _{i}r_{i}^{2}m_{i}}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<munder>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</munder>
<msub>
<mi>I</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>=</mo>
<munder>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</munder>
<msubsup>
<mi>r</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msubsup>
<msub>
<mi>m</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mstyle>
</mrow>
{\displaystyle \sum _{i}I_{i}=\sum _{i}r_{i}^{2}m_{i}}
</semantics>
</math></span><img alt="\sum _{i}I_{i}=\sum _{i}r_{i}^{2}m_{i}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e77d22473cf9495a86e27ee160fcfbd35b50253" style="vertical-align: -3.005ex; width:17.348ex; height:5.509ex;"/></span></dd></dl><p>Angular momentum's dependence on position and shape is reflected in its units versus linear momentum: kg⋅m2/s, N⋅m⋅s, or J⋅s for angular momentum versus kg⋅m/s or N⋅s for linear momentum. When calculating angular momentum as the product of the moment of inertia times the angular velocity, the angular velocity must be expressed in radians per second, where the radian assumes the dimensionless value of unity. (When performing dimensional analysis, it may be productive to use orientational analysis which treats radians as a base unit, but this is outside the scope of the International system of units). Angular momentum's units can be interpreted as torque⋅time or as energy⋅time per angle. An object with angular momentum of <span class="nowrap"><i>L</i> N⋅m⋅s</span> can be reduced to zero rotation (all of the rotational energy can be transferred out of it) by an angular impulse of <span class="nowrap"><i>L</i> N⋅m⋅s</span> or equivalently, by torque or work of <span class="nowrap"><i>L</i> N⋅m</span> for one second, or energy of <span class="nowrap"><i>L</i> J</span> for one second.
</p><p>The plane perpendicular to the axis of angular momentum and passing through the center of mass is sometimes called the <i>invariable plane</i>, because the direction of the axis remains fixed if only the interactions of the bodies within the system, free from outside influences, are considered. One such plane is the invariable plane of the Solar System.
</p><h3><span class="mw-headline" id="Angular_momentum_and_torque">Angular momentum and torque</span><span class="mw-editsection"><span class="mw-editsection-bracket"></span>edit<span class="mw-editsection-bracket"></span></span></h3><p>Newton's second law of motion can be expressed mathematically,
</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle \mathbf {F} =m\mathbf {a} ,}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">F</mi>
</mrow>
<mo>=</mo>
<mi>m</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">a</mi>
</mrow>
<mo>,</mo>
</mstyle>
</mrow>
{\displaystyle \mathbf {F} =m\mathbf {a} ,}
</semantics>
</math></span><img alt="\mathbf {F} =m\mathbf {a} ," aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93aa54e6c7e8df66d85d06b6eb0b0a2d3ec4ce20" style="vertical-align: -0.671ex; width:8.768ex; height:2.509ex;"/></span></dd></dl><p>or force = mass × acceleration. The rotational equivalent for point particles may be derived as follows:
</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle \mathbf {L} =I{\boldsymbol {\omega }}}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">L</mi>
</mrow>
<mo>=</mo>
<mi>I</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold-italic">ω<!-- ω --></mi>
</mrow>
</mstyle>
</mrow>
{\displaystyle \mathbf {L} =I{\boldsymbol {\omega }}}
</semantics>
</math></span><img alt="\mathbf {L} =I{\boldsymbol {\omega }}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/794f5746b0089a4b7de0494195195a6291eda0aa" style="vertical-align: -0.338ex; width:7.547ex; height:2.176ex;"/></span></dd></dl><p>which means that the torque (i.e. the time derivative of the angular momentum) is
</p><dl><dd><span class="mwe-math-element" data-qid="Q48103"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle {\boldsymbol {\tau }}={\frac {dI}{dt}}{\boldsymbol {\omega }}+I{\frac {d{\boldsymbol {\omega }}}{dt}}.}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold-italic">τ<!-- τ --></mi>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>d</mi>
<mi>I</mi>
</mrow>
<mrow>
<mi>d</mi>
<mi>t</mi>
</mrow>
</mfrac>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold-italic">ω<!-- ω --></mi>
</mrow>
<mo>+</mo>
<mi>I</mi>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>d</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold-italic">ω<!-- ω --></mi>
</mrow>
</mrow>
<mrow>
<mi>d</mi>
<mi>t</mi>
</mrow>
</mfrac>
</mrow>
<mo>.</mo>
</mstyle>
</mrow>
{\displaystyle {\boldsymbol {\tau }}={\frac {dI}{dt}}{\boldsymbol {\omega }}+I{\frac {d{\boldsymbol {\omega }}}{dt}}.}
</semantics>
</math></span><img alt="{\displaystyle {\boldsymbol {\tau }}={\frac {dI}{dt}}{\boldsymbol {\omega }}+I{\frac {d{\boldsymbol {\omega }}}{dt}}.}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2c66790aae79872ce38c8f57bb7666a5e5bf05b" style="vertical-align: -2.005ex; width:17.789ex; height:5.509ex;"/></span></dd></dl><p>Because the moment of inertia is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle mr^{2}}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>m</mi>
<msup>
<mi>r</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
{\displaystyle mr^{2}}
</semantics>
</math></span><img alt="mr^{2}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddd9d0ea2911509b014b72a7b536acb7376cb455" style="vertical-align: -0.338ex; width:4.143ex; height:2.676ex;"/></span>, it follows that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle {\frac {dI}{dt}}=2mr{\frac {dr}{dt}}=2rp_{||}}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>d</mi>
<mi>I</mi>
</mrow>
<mrow>
<mi>d</mi>
<mi>t</mi>
</mrow>
</mfrac>
</mrow>
<mo>=</mo>
<mn>2</mn>
<mi>m</mi>
<mi>r</mi>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>d</mi>
<mi>r</mi>
</mrow>
<mrow>
<mi>d</mi>
<mi>t</mi>
</mrow>
</mfrac>
</mrow>
<mo>=</mo>
<mn>2</mn>
<mi>r</mi>
<msub>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
</mrow>
</msub>
</mstyle>
</mrow>
{\displaystyle {\frac {dI}{dt}}=2mr{\frac {dr}{dt}}=2rp_{||}}
</semantics>
</math></span><img alt="{\displaystyle {\frac {dI}{dt}}=2mr{\frac {dr}{dt}}=2rp_{||}}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e7393f1bd48479c3e727ed9bd2a44ba8959f28e" style="vertical-align: -2.005ex; width:21.3ex; height:5.509ex;"/></span>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle {\frac {d\mathbf {L} }{dt}}=I{\frac {d{\boldsymbol {\omega }}}{dt}}+2rp_{||}{\boldsymbol {\omega }},}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>d</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">L</mi>
</mrow>
</mrow>
<mrow>
<mi>d</mi>
<mi>t</mi>
</mrow>
</mfrac>
</mrow>
<mo>=</mo>
<mi>I</mi>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>d</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold-italic">ω<!-- ω --></mi>
</mrow>
</mrow>
<mrow>
<mi>d</mi>
<mi>t</mi>
</mrow>
</mfrac>
</mrow>
<mo>+</mo>
<mn>2</mn>
<mi>r</mi>
<msub>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
</mrow>
</msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold-italic">ω<!-- ω --></mi>
</mrow>
<mo>,</mo>
</mstyle>
</mrow>
{\displaystyle {\frac {d\mathbf {L} }{dt}}=I{\frac {d{\boldsymbol {\omega }}}{dt}}+2rp_{||}{\boldsymbol {\omega }},}
</semantics>
</math></span><img alt="{\displaystyle {\frac {d\mathbf {L} }{dt}}=I{\frac {d{\boldsymbol {\omega }}}{dt}}+2rp_{||}{\boldsymbol {\omega }},}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3fbdc5103e70d211066c7dc4abe45e04df273953" style="vertical-align: -2.005ex; width:21.335ex; height:5.509ex;"/></span> which, reduces to
</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle {\boldsymbol {\tau }}=I{\boldsymbol {\alpha }}+2rp_{||}{\boldsymbol {\omega }}.}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold-italic">τ<!-- τ --></mi>
</mrow>
<mo>=</mo>
<mi>I</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold-italic">α<!-- α --></mi>
</mrow>
<mo>+</mo>
<mn>2</mn>
<mi>r</mi>
<msub>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
</mrow>
</msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold-italic">ω<!-- ω --></mi>
</mrow>
<mo>.</mo>
</mstyle>
</mrow>
{\displaystyle {\boldsymbol {\tau }}=I{\boldsymbol {\alpha }}+2rp_{||}{\boldsymbol {\omega }}.}
</semantics>
</math></span><img alt="{\displaystyle {\boldsymbol {\tau }}=I{\boldsymbol {\alpha }}+2rp_{||}{\boldsymbol {\omega }}.}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7bf36dafd07a96cefd7b472c30ea9b6e1dd4d3bf" style="vertical-align: -1.171ex; width:17.14ex; height:3.009ex;"/></span></dd></dl><p>This is the rotational analog of Newton's Second Law. Note that the torque is not necessarily proportional or parallel to the angular acceleration (as one might expect). The reason for this is that the moment of inertia of a particle can change with time, something that cannot occur for ordinary mass.
</p>samsofihttp://www.blogger.com/profile/12794342593186572847noreply@blogger.com0tag:blogger.com,1999:blog-4126202964944401087.post-60471311418152729682021-01-20T04:07:00.007-08:002021-01-20T04:07:16.665-08:00Conservation of angular momentum<img alt="" class="thumbimage" data-file-height="500" data-file-width="307" decoding="async" height="277" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/68/Cup_of_Russia_2010_-_Yuko_Kawaguti_%282%29.jpg/170px-Cup_of_Russia_2010_-_Yuko_Kawaguti_%282%29.jpg" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/68/Cup_of_Russia_2010_-_Yuko_Kawaguti_%282%29.jpg/255px-Cup_of_Russia_2010_-_Yuko_Kawaguti_%282%29.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/6/68/Cup_of_Russia_2010_-_Yuko_Kawaguti_%282%29.jpg 2x" width="170"/><br/><br/><br/><h3><span class="mw-headline" id="General_considerations">General considerations</span><span class="mw-editsection"><span class="mw-editsection-bracket"></span>edit<span class="mw-editsection-bracket"></span></span></h3><p>A rotational analog of Newton's third law of motion might be written, "In a closed system, no torque can be exerted on any matter without the exertion on some other matter of an equal and opposite torque." Hence, <i>angular momentum can be exchanged between objects in a closed system, but total angular momentum before and after an exchange remains constant (is conserved).</i>
</p><p>Seen another way, a rotational analogue of Newton's first law of motion might be written, "A rigid body continues in a state of uniform rotation unless acted by an external influence." Thus <i>with no external influence to act upon it, the original angular momentum of the system remains constant</i>.
</p><p>The conservation of angular momentum is used in analyzing <i>central force motion</i>. If the net force on some body is directed always toward some point, the <i>center</i>, then there is no torque on the body with respect to the center, as all of the force is directed along the radius vector, and none is perpendicular to the radius. Mathematically, torque <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle {\boldsymbol {\tau }}=\mathbf {r} \times \mathbf {F} =\mathbf {0} ,}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold-italic">τ<!-- τ --></mi>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">r</mi>
</mrow>
<mo>×<!-- × --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">F</mi>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn mathvariant="bold">0</mn>
</mrow>
<mo>,</mo>
</mstyle>
</mrow>
{\displaystyle {\boldsymbol {\tau }}=\mathbf {r} \times \mathbf {F} =\mathbf {0} ,}
</semantics>
</math></span><img alt="{\boldsymbol {\tau }}=\mathbf {r} \times \mathbf {F} =\mathbf {0} ," aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bce23213f7078a404609088e6b314db820290527" style="vertical-align: -0.671ex; width:15.224ex; height:2.509ex;"/></span> because in this case <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle \mathbf {r} }" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">r</mi>
</mrow>
</mstyle>
</mrow>
{\displaystyle \mathbf {r} }
</semantics>
</math></span><img alt="\mathbf {r} " aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eca0f46511c4c986c48b254073732c0bd98ae0c1" style="vertical-align: -0.338ex; width:1.102ex; height:1.676ex;"/></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle \mathbf {F} }" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">F</mi>
</mrow>
</mstyle>
</mrow>
{\displaystyle \mathbf {F} }
</semantics>
</math></span><img alt="\mathbf {F} " aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da18bef8c979f3548bb0d8976f5844012d7b8256" style="vertical-align: -0.338ex; width:1.683ex; height:2.176ex;"/></span> are parallel vectors. Therefore, the angular momentum of the body about the center is constant. This is the case with gravitational attraction in the orbits of planets and satellites, where the gravitational force is always directed toward the primary body and orbiting bodies conserve angular momentum by exchanging distance and velocity as they move about the primary. Central force motion is also used in the analysis of the Bohr model of the atom.
</p><p>For a planet, angular momentum is distributed between the spin of the planet and its revolution in its orbit, and these are often exchanged by various mechanisms. The conservation of angular momentum in the Earth–Moon system results in the transfer of angular momentum from Earth to Moon, due to tidal torque the Moon exerts on the Earth. This in turn results in the slowing down of the rotation rate of Earth, at about 65.7 nanoseconds per day, and in gradual increase of the radius of Moon's orbit, at about 3.82 centimeters per year.
</p><p>The conservation of angular momentum explains the angular acceleration of an ice skater as she brings her arms and legs close to the vertical axis of rotation. By bringing part of the mass of her body closer to the axis, she decreases her body's moment of inertia. Because angular momentum is the product of moment of inertia and angular velocity, if the angular momentum remains constant (is conserved), then the angular velocity (rotational speed) of the skater must increase.
</p><p>The same phenomenon results in extremely fast spin of compact stars (like white dwarfs, neutron stars and black holes) when they are formed out of much larger and slower rotating stars. Decrease in the size of an object <i>n</i> times results in increase of its angular velocity by the factor of <i>n</i>2.
</p><p>Conservation is not always a full explanation for the dynamics of a system but is a key constraint. For example, a spinning top is subject to gravitational torque making it lean over and change the angular momentum about the nutation axis, but neglecting friction at the point of spinning contact, it has a conserved angular momentum about its spinning axis, and another about its precession axis. Also, in any planetary system, the planets, star(s), comets, and asteroids can all move in numerous complicated ways, but only so that the angular momentum of the system is conserved.
</p><p>Noether's theorem states that every conservation law is associated with a symmetry (invariant) of the underlying physics. The symmetry associated with conservation of angular momentum is rotational invariance. The fact that the physics of a system is unchanged if it is rotated by any angle about an axis implies that angular momentum is conserved.
</p><h3><span id="Relation_to_Newton.27s_second_law_of_motion"></span><span class="mw-headline" id="Relation_to_Newton's_second_law_of_motion">Relation to Newton's second law of motion</span><span class="mw-editsection"><span class="mw-editsection-bracket"></span>edit<span class="mw-editsection-bracket"></span></span></h3><p>While angular momentum total conservation can be understood separately from Newton's laws of motion as stemming from Noether's theorem in systems symmetric under rotations, it can also be understood simply as an efficient method of calculation of results that can also be otherwise arrived at directly from Newton's second law, together with laws governing the forces of nature (such as Newton's third law, Maxwell's equations and Lorentz force). Indeed, given initial conditions of position and velocity for every point, and the forces at such a condition, one may use Newton's second law to calculate the second derivative of position, and solving for this gives full information on the development of the physical system with time. Note, however, that this is no longer true in quantum mechanics, due to the existence of particle spin, which is angular momentum that cannot be described by the cumulative effect of point-like motions in space.
</p><p>As an example, consider decreasing of the moment of inertia, e.g. when a figure skater is pulling in her/his hands, speeding up the circular motion. In terms of angular momentum conservation, we have, for angular momentum <i>L</i>, moment of inertia <i>I</i> and angular velocity <i>ω</i>:
</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle 0=dL=d(I\cdot \omega )=dI\cdot \omega +I\cdot d\omega }" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>0</mn>
<mo>=</mo>
<mi>d</mi>
<mi>L</mi>
<mo>=</mo>
<mi>d</mi>
<mo stretchy="false">(</mo>
<mi>I</mi>
<mo>⋅<!-- ⋅ --></mo>
<mi>ω<!-- ω --></mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>d</mi>
<mi>I</mi>
<mo>⋅<!-- ⋅ --></mo>
<mi>ω<!-- ω --></mi>
<mo>+</mo>
<mi>I</mi>
<mo>⋅<!-- ⋅ --></mo>
<mi>d</mi>
<mi>ω<!-- ω --></mi>
</mstyle>
</mrow>
{\displaystyle 0=dL=d(I\cdot \omega )=dI\cdot \omega +I\cdot d\omega }
</semantics>
</math></span><img alt="{\displaystyle 0=dL=d(I\cdot \omega )=dI\cdot \omega +I\cdot d\omega }" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d15093607ae440e94bbf63e759eb62b687ab1110" style="vertical-align: -0.838ex; width:34.444ex; height:2.843ex;"/></span></dd></dl><p>Using this, we see that the change requires an energy of:
</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle dE=d\left({\frac {1}{2}}I\cdot \omega ^{2}\right)={\frac {1}{2}}dI\cdot \omega ^{2}+I\cdot \omega \cdot d\omega =-{\frac {1}{2}}dI\cdot \omega ^{2}}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>d</mi>
<mi>E</mi>
<mo>=</mo>
<mi>d</mi>
<mrow>
<mo>(</mo>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
<mi>I</mi>
<mo>⋅<!-- ⋅ --></mo>
<msup>
<mi>ω<!-- ω --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
<mi>d</mi>
<mi>I</mi>
<mo>⋅<!-- ⋅ --></mo>
<msup>
<mi>ω<!-- ω --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>+</mo>
<mi>I</mi>
<mo>⋅<!-- ⋅ --></mo>
<mi>ω<!-- ω --></mi>
<mo>⋅<!-- ⋅ --></mo>
<mi>d</mi>
<mi>ω<!-- ω --></mi>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
<mi>d</mi>
<mi>I</mi>
<mo>⋅<!-- ⋅ --></mo>
<msup>
<mi>ω<!-- ω --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
{\displaystyle dE=d\left({\frac {1}{2}}I\cdot \omega ^{2}\right)={\frac {1}{2}}dI\cdot \omega ^{2}+I\cdot \omega \cdot d\omega =-{\frac {1}{2}}dI\cdot \omega ^{2}}
</semantics>
</math></span><img alt="{\displaystyle dE=d\left({\frac {1}{2}}I\cdot \omega ^{2}\right)={\frac {1}{2}}dI\cdot \omega ^{2}+I\cdot \omega \cdot d\omega =-{\frac {1}{2}}dI\cdot \omega ^{2}}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/619bc45463c048a54adf9aca6586508e8caf02e8" style="vertical-align: -2.505ex; width:55.077ex; height:6.176ex;"/></span></dd></dl><p>so that a decrease in the moment of inertia requires investing energy.
</p><p>This can be compared to the work done as calculated using Newton's laws. Each point in the rotating body is accelerating, at each point of time, with radial acceleration of:
</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle -r\cdot \omega ^{2}}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo>−<!-- − --></mo>
<mi>r</mi>
<mo>⋅<!-- ⋅ --></mo>
<msup>
<mi>ω<!-- ω --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
{\displaystyle -r\cdot \omega ^{2}}
</semantics>
</math></span><img alt="{\displaystyle -r\cdot \omega ^{2}}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/988f476c4a01afb3d9d6b6d6aba262632eeebea9" style="vertical-align: -0.505ex; width:7.036ex; height:2.843ex;"/></span></dd></dl><p>Let us observe a point of mass <i>m</i>, whose position vector relative to the center of motion is parallel to the z-axis at a given point of time, and is at a distance <i>z</i>. The centripetal force on this point, keeping the circular motion, is:
</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle -m\cdot z\cdot \omega ^{2}}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo>−<!-- − --></mo>
<mi>m</mi>
<mo>⋅<!-- ⋅ --></mo>
<mi>z</mi>
<mo>⋅<!-- ⋅ --></mo>
<msup>
<mi>ω<!-- ω --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
{\displaystyle -m\cdot z\cdot \omega ^{2}}
</semantics>
</math></span><img alt="{\displaystyle -m\cdot z\cdot \omega ^{2}}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5b5d5e9aee971b197b897a9c6f8c1e41968260d" style="vertical-align: -0.505ex; width:10.795ex; height:2.843ex;"/></span></dd></dl><p>Thus the work required for moving this point to a distance <i>dz</i> farther from the center of motion is:
</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle dW=-m\cdot z\cdot \omega ^{2}\cdot dz=-m\cdot \omega ^{2}\cdot d\left({\frac {1}{2}}z^{2}\right)}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>d</mi>
<mi>W</mi>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mi>m</mi>
<mo>⋅<!-- ⋅ --></mo>
<mi>z</mi>
<mo>⋅<!-- ⋅ --></mo>
<msup>
<mi>ω<!-- ω --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>⋅<!-- ⋅ --></mo>
<mi>d</mi>
<mi>z</mi>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mi>m</mi>
<mo>⋅<!-- ⋅ --></mo>
<msup>
<mi>ω<!-- ω --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>⋅<!-- ⋅ --></mo>
<mi>d</mi>
<mrow>
<mo>(</mo>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
<msup>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
<mo>)</mo>
</mrow>
</mstyle>
</mrow>
{\displaystyle dW=-m\cdot z\cdot \omega ^{2}\cdot dz=-m\cdot \omega ^{2}\cdot d\left({\frac {1}{2}}z^{2}\right)}
</semantics>
</math></span><img alt="{\displaystyle dW=-m\cdot z\cdot \omega ^{2}\cdot dz=-m\cdot \omega ^{2}\cdot d\left({\frac {1}{2}}z^{2}\right)}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba60f4f0499b41a546eff2437ee1cdabc0c6736b" style="vertical-align: -2.505ex; width:43.5ex; height:6.176ex;"/></span></dd></dl><p>For a non-pointlike body one must integrate over this, with <i>m</i> replaced by the mass density per unit <i>z</i>. This gives:
</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle dW=-{\frac {1}{2}}dI\cdot \omega ^{2}}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>d</mi>
<mi>W</mi>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
<mi>d</mi>
<mi>I</mi>
<mo>⋅<!-- ⋅ --></mo>
<msup>
<mi>ω<!-- ω --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
{\displaystyle dW=-{\frac {1}{2}}dI\cdot \omega ^{2}}
</semantics>
</math></span><img alt="{\displaystyle dW=-{\frac {1}{2}}dI\cdot \omega ^{2}}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/081f58278e2a35bdb47345ef6de64876943d2fa7" style="vertical-align: -1.838ex; width:17.123ex; height:5.176ex;"/></span></dd></dl><p>which is exactly the energy required for keeping the angular momentum conserved.
</p><p>Note, that the above calculation can also be performed per mass, using kinematics only. Thus the phenomena of figure skater accelerating tangential velocity while pulling her/his hands in, can be understood as follows in layman's language: The skater's palms are not moving in a straight line, so they are constantly accelerating inwards, but do not gain additional speed because the accelerating is always done when their motion inwards is zero. However, this is different when pulling the palms closer to the body: The acceleration due to rotation now increases the speed; but because of the rotation, the increase in speed does not translate to a significant speed inwards, but to an increase of the rotation speed.
</p><h3><span class="mw-headline" id="In_Lagrangian_formalism">In Lagrangian formalism</span><span class="mw-editsection"><span class="mw-editsection-bracket"></span>edit<span class="mw-editsection-bracket"></span></span></h3><p>In Lagrangian mechanics, angular momentum for rotation around a given axis, is the conjugate momentum of the generalized coordinate of the angle around the same axis. For example, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle L_{z}}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>L</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>z</mi>
</mrow>
</msub>
</mstyle>
</mrow>
{\displaystyle L_{z}}
</semantics>
</math></span><img alt="L_{z}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77a5b940110c5e1fe03782a31c5e700939ae20e6" style="vertical-align: -0.671ex; width:2.585ex; height:2.509ex;"/></span>, the angular momentum around the z axis, is:
</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle L_{z}={\frac {\partial {\cal {L}}}{\partial {\dot {\theta }}_{z}}}}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>L</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>z</mi>
</mrow>
</msub>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">L</mi>
</mrow>
</mrow>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
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<mi>θ<!-- θ --></mi>
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<mrow class="MJX-TeXAtom-ORD">
<mi>z</mi>
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</msub>
</mrow>
</mfrac>
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{\displaystyle L_{z}={\frac {\partial {\cal {L}}}{\partial {\dot {\theta }}_{z}}}}
</semantics>
</math></span><img alt="{\displaystyle L_{z}={\frac {\partial {\cal {L}}}{\partial {\dot {\theta }}_{z}}}}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46ec2a2d5043dc0be22545107c74c2a5d9e703a3" style="vertical-align: -2.838ex; width:10.195ex; height:6.343ex;"/></span></dd></dl><p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle {\cal {L}}}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">L</mi>
</mrow>
</mrow>
</mstyle>
</mrow>
{\displaystyle {\cal {L}}}
</semantics>
</math></span><img alt="{\displaystyle {\cal {L}}}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cddbb21ad79aa4e70f27927e433fd985873a3b6a" style="vertical-align: -0.338ex; width:1.604ex; height:2.176ex;"/></span> is the Lagrangian and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle \theta _{z}}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>θ<!-- θ --></mi>
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<mi>z</mi>
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{\displaystyle \theta _{z}}
</semantics>
</math></span><img alt="{\displaystyle \theta _{z}}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/043cbe96ac7ea3ea664c287abee4ea1819373353" style="vertical-align: -0.671ex; width:2.092ex; height:2.509ex;"/></span> is the angle around the z axis.
</p><p>Note that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle {\dot {\theta }}_{z}}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>θ<!-- θ --></mi>
<mo>˙<!-- ˙ --></mo>
</mover>
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</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>z</mi>
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{\displaystyle {\dot {\theta }}_{z}}
</semantics>
</math></span><img alt="{\displaystyle {\dot {\theta }}_{z}}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebb1c138bd0e051666aae97e19177d0f45976d30" style="vertical-align: -0.671ex; width:2.358ex; height:3.176ex;"/></span>, the time derivative of the angle, is the angular velocity <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle \omega _{z}}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>ω<!-- ω --></mi>
<mrow class="MJX-TeXAtom-ORD">
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{\displaystyle \omega _{z}}
</semantics>
</math></span><img alt="\omega _{z}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6664a519a737b4e06b4668df3e837d4e4986db1" style="vertical-align: -0.671ex; width:2.447ex; height:2.009ex;"/></span>. Ordinarily, the Lagrangian depends on the angular velocity through the kinetic energy: The latter can be written by separating the velocity to its radial and tangential part, with the tangential part at the x-y plane, around the z-axis, being equal to:
</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle \sum _{i}{\frac {1}{2}}m_{i}{v_{T}}_{i}^{2}=\sum _{i}{\frac {1}{2}}m_{i}(x_{i}^{2}+y_{i}^{2}){{\omega _{z}}_{i}}^{2}}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
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<munder>
<mo>∑<!-- ∑ --></mo>
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<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
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<mo>=</mo>
<munder>
<mo>∑<!-- ∑ --></mo>
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<mi>i</mi>
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</munder>
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<msub>
<mi>m</mi>
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<mo stretchy="false">(</mo>
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<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
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<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
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</msubsup>
<mo>+</mo>
<msubsup>
<mi>y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msubsup>
<mo stretchy="false">)</mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>ω<!-- ω --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>z</mi>
</mrow>
</msub>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
{\displaystyle \sum _{i}{\frac {1}{2}}m_{i}{v_{T}}_{i}^{2}=\sum _{i}{\frac {1}{2}}m_{i}(x_{i}^{2}+y_{i}^{2}){{\omega _{z}}_{i}}^{2}}
</semantics>
</math></span><img alt="{\displaystyle \sum _{i}{\frac {1}{2}}m_{i}{v_{T}}_{i}^{2}=\sum _{i}{\frac {1}{2}}m_{i}(x_{i}^{2}+y_{i}^{2}){{\omega _{z}}_{i}}^{2}}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cee1659bc70ac90e0055ab40b6483eaff055d5ca" style="vertical-align: -3.005ex; width:37.381ex; height:6.343ex;"/></span></dd></dl><p>where the subscript i stands for the i-th body, and <i>m</i>, <i>v</i><sub><i>T</i></sub> and <i>ω</i><sub><i>z</i></sub> stand for mass, tangential velocity around the z-axis and angular velocity around that axis, respectively.
</p><p>For a body that is not point-like, with density <i>ρ</i>, we have instead:
</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle {\frac {1}{2}}\int \rho (x,y,z)(x_{i}^{2}+y_{i}^{2}){{\omega _{z}}_{i}}^{2}={\frac {1}{2}}{I_{z}}_{i}{{\omega _{z}}_{i}}^{2}}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
<mo>∫<!-- ∫ --></mo>
<mi>ρ<!-- ρ --></mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>,</mo>
<mi>y</mi>
<mo>,</mo>
<mi>z</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<msubsup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msubsup>
<mo>+</mo>
<msubsup>
<mi>y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msubsup>
<mo stretchy="false">)</mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>ω<!-- ω --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>z</mi>
</mrow>
</msub>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>I</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>z</mi>
</mrow>
</msub>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>ω<!-- ω --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>z</mi>
</mrow>
</msub>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
{\displaystyle {\frac {1}{2}}\int \rho (x,y,z)(x_{i}^{2}+y_{i}^{2}){{\omega _{z}}_{i}}^{2}={\frac {1}{2}}{I_{z}}_{i}{{\omega _{z}}_{i}}^{2}}
</semantics>
</math></span><img alt="{\displaystyle {\frac {1}{2}}\int \rho (x,y,z)(x_{i}^{2}+y_{i}^{2}){{\omega _{z}}_{i}}^{2}={\frac {1}{2}}{I_{z}}_{i}{{\omega _{z}}_{i}}^{2}}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9575a031ca5b31e0509754eca60dd1fa337e3c8a" style="vertical-align: -2.338ex; width:39.392ex; height:5.676ex;"/></span></dd></dl><p>where <i>I</i><sub>z</sub> is the moment of inertia around the z-axis.
</p><p>Thus, assuming the potential energy does not depend on <i>ω</i><sub><i>z</i></sub> (this assumption may fail for electromagnetic systems), we have the angular momentum of the i-th object:
</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle {\begin{aligned}{L_{z}}_{i}&={\frac {\partial {\cal {L}}}{\partial {{\omega _{z}}_{i}}}}={\frac {\partial E_{k}}{\partial {{\omega _{z}}_{i}}}}\\&={I_{z}}_{i}\cdot {\omega _{z}}_{i}\end{aligned}}}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mtable columnalign="right left right left right left right left right left right left" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true" rowspacing="3pt">
<mtr>
<mtd>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>L</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>z</mi>
</mrow>
</msub>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">L</mi>
</mrow>
</mrow>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>ω<!-- ω --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>z</mi>
</mrow>
</msub>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mrow>
</mrow>
</mfrac>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<msub>
<mi>E</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>ω<!-- ω --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>z</mi>
</mrow>
</msub>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mrow>
</mrow>
</mfrac>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd></mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>I</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>z</mi>
</mrow>
</msub>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>⋅<!-- ⋅ --></mo>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>ω<!-- ω --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>z</mi>
</mrow>
</msub>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mtd>
</mtr>
</mtable>
</mrow>
</mstyle>
</mrow>
{\displaystyle {\begin{aligned}{L_{z}}_{i}&={\frac {\partial {\cal {L}}}{\partial {{\omega _{z}}_{i}}}}={\frac {\partial E_{k}}{\partial {{\omega _{z}}_{i}}}}\\&={I_{z}}_{i}\cdot {\omega _{z}}_{i}\end{aligned}}}
</semantics>
</math></span><img alt="{\displaystyle {\begin{aligned}{L_{z}}_{i}&={\frac {\partial {\cal {L}}}{\partial {{\omega _{z}}_{i}}}}={\frac {\partial E_{k}}{\partial {{\omega _{z}}_{i}}}}\\&={I_{z}}_{i}\cdot {\omega _{z}}_{i}\end{aligned}}}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ced48449127d30dc5fcde4c5eab09328b68c93b9" style="vertical-align: -3.838ex; width:21.135ex; height:8.843ex;"/></span></dd></dl><p>We have thus far rotated each object by a separate angle; we may also define an overall angle <i>θ</i><sub>z</sub> by which we rotate the whole system, thus rotating also each object around the z-axis, and have the overall angular momentum:
</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle L_{z}=\sum _{i}{I_{z}}_{i}\cdot {\omega _{z}}_{i}}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>L</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>z</mi>
</mrow>
</msub>
<mo>=</mo>
<munder>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</munder>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>I</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>z</mi>
</mrow>
</msub>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>⋅<!-- ⋅ --></mo>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>ω<!-- ω --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>z</mi>
</mrow>
</msub>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mstyle>
</mrow>
{\displaystyle L_{z}=\sum _{i}{I_{z}}_{i}\cdot {\omega _{z}}_{i}}
</semantics>
</math></span><img alt="{\displaystyle L_{z}=\sum _{i}{I_{z}}_{i}\cdot {\omega _{z}}_{i}}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15e4a01264119c9ff46f2e4099dcc92eec496a41" style="vertical-align: -3.005ex; width:17.176ex; height:5.509ex;"/></span></dd></dl><p>From Euler-Lagrange equations it then follows that:
</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle 0={\frac {\partial {\cal {L}}}{\partial {{\theta _{z}}_{i}}}}-{\frac {d}{dt}}\left({\frac {\partial {\cal {L}}}{\partial {{{\dot {\theta }}_{z}}_{i}}}}\right)={\frac {\partial {\cal {L}}}{\partial {{\theta _{z}}_{i}}}}-{\frac {d{L_{z}}_{i}}{dt}}}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>0</mn>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">L</mi>
</mrow>
</mrow>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>θ<!-- θ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>z</mi>
</mrow>
</msub>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mrow>
</mrow>
</mfrac>
</mrow>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>d</mi>
<mrow>
<mi>d</mi>
<mi>t</mi>
</mrow>
</mfrac>
</mrow>
<mrow>
<mo>(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">L</mi>
</mrow>
</mrow>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>θ<!-- θ --></mi>
<mo>˙<!-- ˙ --></mo>
</mover>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>z</mi>
</mrow>
</msub>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mrow>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">L</mi>
</mrow>
</mrow>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>θ<!-- θ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>z</mi>
</mrow>
</msub>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mrow>
</mrow>
</mfrac>
</mrow>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>d</mi>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>L</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>z</mi>
</mrow>
</msub>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mrow>
<mrow>
<mi>d</mi>
<mi>t</mi>
</mrow>
</mfrac>
</mrow>
</mstyle>
</mrow>
{\displaystyle 0={\frac {\partial {\cal {L}}}{\partial {{\theta _{z}}_{i}}}}-{\frac {d}{dt}}\left({\frac {\partial {\cal {L}}}{\partial {{{\dot {\theta }}_{z}}_{i}}}}\right)={\frac {\partial {\cal {L}}}{\partial {{\theta _{z}}_{i}}}}-{\frac {d{L_{z}}_{i}}{dt}}}
</semantics>
</math></span><img alt="{\displaystyle 0={\frac {\partial {\cal {L}}}{\partial {{\theta _{z}}_{i}}}}-{\frac {d}{dt}}\left({\frac {\partial {\cal {L}}}{\partial {{{\dot {\theta }}_{z}}_{i}}}}\right)={\frac {\partial {\cal {L}}}{\partial {{\theta _{z}}_{i}}}}-{\frac {d{L_{z}}_{i}}{dt}}}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4eca9ec00bbbf83969b41733aaa004b16f23484" style="vertical-align: -3.171ex; width:40.84ex; height:7.509ex;"/></span></dd></dl><p>Since the lagrangian is dependent upon the angles of the object only through the potential, we have:
</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle {\frac {d{L_{z}}_{i}}{dt}}={\frac {\partial {\cal {L}}}{\partial {{\theta _{z}}_{i}}}}=-{\frac {\partial V}{\partial {{\theta _{z}}_{i}}}}}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>d</mi>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>L</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>z</mi>
</mrow>
</msub>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mrow>
<mrow>
<mi>d</mi>
<mi>t</mi>
</mrow>
</mfrac>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">L</mi>
</mrow>
</mrow>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>θ<!-- θ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>z</mi>
</mrow>
</msub>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mrow>
</mrow>
</mfrac>
</mrow>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>V</mi>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>θ<!-- θ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>z</mi>
</mrow>
</msub>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mrow>
</mrow>
</mfrac>
</mrow>
</mstyle>
</mrow>
{\displaystyle {\frac {d{L_{z}}_{i}}{dt}}={\frac {\partial {\cal {L}}}{\partial {{\theta _{z}}_{i}}}}=-{\frac {\partial V}{\partial {{\theta _{z}}_{i}}}}}
</semantics>
</math></span><img alt="{\displaystyle {\frac {d{L_{z}}_{i}}{dt}}={\frac {\partial {\cal {L}}}{\partial {{\theta _{z}}_{i}}}}=-{\frac {\partial V}{\partial {{\theta _{z}}_{i}}}}}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e200f643c16a72c101334aea3c7b2fcb031059e1" style="vertical-align: -2.505ex; width:23.533ex; height:6.009ex;"/></span></dd></dl><p>which is the torque on the i-th object.
</p><p>Suppose the system is invariant to rotations, so that the potential is independent of an overall rotation by the angle <i>θ</i><sub>z</sub> (thus it may depend on the angles of objects only through their differences, in the form <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle V({\theta _{z}}_{i},{\theta _{z}}_{j})=V({\theta _{z}}_{i}-{\theta _{z}}_{j})}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>V</mi>
<mo stretchy="false">(</mo>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>θ<!-- θ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>z</mi>
</mrow>
</msub>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>θ<!-- θ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>z</mi>
</mrow>
</msub>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>V</mi>
<mo stretchy="false">(</mo>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>θ<!-- θ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>z</mi>
</mrow>
</msub>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>−<!-- − --></mo>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>θ<!-- θ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>z</mi>
</mrow>
</msub>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
{\displaystyle V({\theta _{z}}_{i},{\theta _{z}}_{j})=V({\theta _{z}}_{i}-{\theta _{z}}_{j})}
</semantics>
</math></span><img alt="{\displaystyle V({\theta _{z}}_{i},{\theta _{z}}_{j})=V({\theta _{z}}_{i}-{\theta _{z}}_{j})}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bdf948ed90f5ec985945fc05ff41552e955811fb" style="vertical-align: -1.005ex; width:25.953ex; height:3.009ex;"/></span>). We therefore get for the total angular momentum:
</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle {\frac {dL_{z}}{dt}}=-{\frac {\partial V}{\partial {\theta _{z}}}}=0}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>d</mi>
<msub>
<mi>L</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>z</mi>
</mrow>
</msub>
</mrow>
<mrow>
<mi>d</mi>
<mi>t</mi>
</mrow>
</mfrac>
</mrow>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>V</mi>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>θ<!-- θ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>z</mi>
</mrow>
</msub>
</mrow>
</mrow>
</mfrac>
</mrow>
<mo>=</mo>
<mn>0</mn>
</mstyle>
</mrow>
{\displaystyle {\frac {dL_{z}}{dt}}=-{\frac {\partial V}{\partial {\theta _{z}}}}=0}
</semantics>
</math></span><img alt="{\displaystyle {\frac {dL_{z}}{dt}}=-{\frac {\partial V}{\partial {\theta _{z}}}}=0}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aadf954d3dc9b7f87b433a5420606bc0e0fc27b9" style="vertical-align: -2.338ex; width:18.05ex; height:5.843ex;"/></span></dd></dl><p>And thus the angular momentum around the z-axis is conserved.
</p><p>This analysis can be repeated separately for each axis, giving conversation of the angular momentum vector. However, the angles around the three axes cannot be treated simultaneously as generalized coordinates, since they are not independent; in particular, two angles per point suffice to determine its position. While it is true that in the case of a rigid body, fully describing it requires, in addition to three translational degrees of freedom, also specification of three rotational degrees of freedom; however these cannot be defined as rotations around the Cartesian axes (see Euler angles). This caveat is reflected in quantum mechanics in the non-trivial commutation relations of the different components of the angular momentum operator.
</p><h3><span class="mw-headline" id="In_Hamiltonian_formalism">In Hamiltonian formalism</span><span class="mw-editsection"><span class="mw-editsection-bracket"></span>edit<span class="mw-editsection-bracket"></span></span></h3><p>Equivalently, in Hamiltonian mechanics the Hamiltonian can be described as a function of the angular momentum. As before, the part of the kinetic energy related to rotation around the z-axis for the i-th object is:
</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle {\frac {1}{2}}{I_{z}}_{i}{{\omega _{z}}_{i}}^{2}={\frac {{{L_{z}}_{i}}^{2}}{2{I_{z}}_{i}}}}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>I</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>z</mi>
</mrow>
</msub>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>ω<!-- ω --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>z</mi>
</mrow>
</msub>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>L</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>z</mi>
</mrow>
</msub>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mrow>
<mn>2</mn>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>I</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>z</mi>
</mrow>
</msub>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mrow>
</mfrac>
</mrow>
</mstyle>
</mrow>
{\displaystyle {\frac {1}{2}}{I_{z}}_{i}{{\omega _{z}}_{i}}^{2}={\frac {{{L_{z}}_{i}}^{2}}{2{I_{z}}_{i}}}}
</semantics>
</math></span><img alt="{\displaystyle {\frac {1}{2}}{I_{z}}_{i}{{\omega _{z}}_{i}}^{2}={\frac {{{L_{z}}_{i}}^{2}}{2{I_{z}}_{i}}}}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fea4670d6d67a9c53d3b14e287afe4b053b0de0" style="vertical-align: -2.338ex; width:17.498ex; height:6.343ex;"/></span></dd></dl><p>which is analogous to the energy dependence upon momentum along the z-axis, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle {\frac {{{p_{z}}_{i}}^{2}}{{2m}_{i}}}}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>z</mi>
</mrow>
</msub>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
<mi>m</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mfrac>
</mrow>
</mstyle>
</mrow>
{\displaystyle {\frac {{{p_{z}}_{i}}^{2}}{{2m}_{i}}}}
</semantics>
</math></span><img alt="{\displaystyle {\frac {{{p_{z}}_{i}}^{2}}{{2m}_{i}}}}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a130e7f0e2bb096205ed92293b706638b0422cb2" style="vertical-align: -2.171ex; width:4.861ex; height:6.176ex;"/></span>.
</p><p>Hamilton's equations relate the angle around the z-axis to its conjugate momentum, the angular momentum around the same axis:
</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle {\begin{aligned}{\frac {d{\theta _{z}}_{i}}{dt}}&={\frac {\partial {\cal {H}}}{\partial {L_{z}}_{i}}}={\frac {{L_{z}}_{i}}{{I_{z}}_{i}}}\\{\frac {d{L_{z}}_{i}}{dt}}&=-{\frac {\partial {\cal {H}}}{\partial {\theta _{z}}_{i}}}=-{\frac {\partial V}{\partial {\theta _{z}}_{i}}}\end{aligned}}}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mtable columnalign="right left right left right left right left right left right left" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true" rowspacing="3pt">
<mtr>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>d</mi>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>θ<!-- θ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>z</mi>
</mrow>
</msub>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mrow>
<mrow>
<mi>d</mi>
<mi>t</mi>
</mrow>
</mfrac>
</mrow>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">H</mi>
</mrow>
</mrow>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>L</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>z</mi>
</mrow>
</msub>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mrow>
</mfrac>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>L</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>z</mi>
</mrow>
</msub>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>I</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>z</mi>
</mrow>
</msub>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mfrac>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>d</mi>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>L</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>z</mi>
</mrow>
</msub>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mrow>
<mrow>
<mi>d</mi>
<mi>t</mi>
</mrow>
</mfrac>
</mrow>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">H</mi>
</mrow>
</mrow>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>θ<!-- θ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>z</mi>
</mrow>
</msub>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mrow>
</mfrac>
</mrow>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>V</mi>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>θ<!-- θ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>z</mi>
</mrow>
</msub>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mrow>
</mfrac>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
</mstyle>
</mrow>
{\displaystyle {\begin{aligned}{\frac {d{\theta _{z}}_{i}}{dt}}&={\frac {\partial {\cal {H}}}{\partial {L_{z}}_{i}}}={\frac {{L_{z}}_{i}}{{I_{z}}_{i}}}\\{\frac {d{L_{z}}_{i}}{dt}}&=-{\frac {\partial {\cal {H}}}{\partial {\theta _{z}}_{i}}}=-{\frac {\partial V}{\partial {\theta _{z}}_{i}}}\end{aligned}}}
</semantics>
</math></span><img alt="{\displaystyle {\begin{aligned}{\frac {d{\theta _{z}}_{i}}{dt}}&={\frac {\partial {\cal {H}}}{\partial {L_{z}}_{i}}}={\frac {{L_{z}}_{i}}{{I_{z}}_{i}}}\\{\frac {d{L_{z}}_{i}}{dt}}&=-{\frac {\partial {\cal {H}}}{\partial {\theta _{z}}_{i}}}=-{\frac {\partial V}{\partial {\theta _{z}}_{i}}}\end{aligned}}}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de240d5a7cc61a43c85a21401a23d8ae1d66f2a4" style="vertical-align: -5.505ex; width:26.093ex; height:12.176ex;"/></span></dd></dl><p>The first equation gives
</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle {L_{z}}_{i}={I_{z}}_{i}\cdot {{{\dot {\theta }}_{z}}_{i}}={I_{z}}_{i}\cdot {\omega _{z}}_{i}}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>L</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>z</mi>
</mrow>
</msub>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>=</mo>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>I</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>z</mi>
</mrow>
</msub>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>⋅<!-- ⋅ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>θ<!-- θ --></mi>
<mo>˙<!-- ˙ --></mo>
</mover>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>z</mi>
</mrow>
</msub>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mrow>
<mo>=</mo>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>I</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>z</mi>
</mrow>
</msub>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>⋅<!-- ⋅ --></mo>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>ω<!-- ω --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>z</mi>
</mrow>
</msub>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mstyle>
</mrow>
{\displaystyle {L_{z}}_{i}={I_{z}}_{i}\cdot {{{\dot {\theta }}_{z}}_{i}}={I_{z}}_{i}\cdot {\omega _{z}}_{i}}
</semantics>
</math></span><img alt="{\displaystyle {L_{z}}_{i}={I_{z}}_{i}\cdot {{{\dot {\theta }}_{z}}_{i}}={I_{z}}_{i}\cdot {\omega _{z}}_{i}}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8fff0339cc32d6f14853026e32bfeaa291018302" style="vertical-align: -0.671ex; width:24.993ex; height:3.176ex;"/></span></dd></dl><p>And so we get the same results as in the Lagrangian formalism.
</p><p>Note, that for combining all axes together, we write the kinetic energy as:
</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle E_{k}={\frac {1}{2}}\sum _{i}{\frac {|{\bf {{p}_{i}|^{2}}}}{2m_{i}}}=\sum _{i}\left({\frac {{{p_{r}}_{i}}^{2}}{2m_{i}}}+{\frac {1}{2}}{\bf {{L}_{i}}}^{\textsf {T}}{I_{i}}^{-1}{\bf {{L}_{i}}}\right)}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>E</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
<munder>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</munder>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
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<mo stretchy="false">|</mo>
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<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">p</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">i</mi>
</mrow>
</msub>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mo mathvariant="bold" stretchy="false">|</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn mathvariant="bold">2</mn>
</mrow>
</msup>
</mrow>
</mrow>
</mrow>
<mrow>
<mn>2</mn>
<msub>
<mi>m</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mrow>
</mfrac>
</mrow>
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<munder>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
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</mrow>
</munder>
<mrow>
<mo>(</mo>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
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<mrow class="MJX-TeXAtom-ORD">
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<mrow class="MJX-TeXAtom-ORD">
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</msub>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
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</mrow>
</msub>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
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<msub>
<mi>m</mi>
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</mrow>
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</mrow>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
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<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">L</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
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</mrow>
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<mrow class="MJX-TeXAtom-ORD">
<mtext mathvariant="sans-serif">T</mtext>
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</mrow>
</msub>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<msub>
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<mrow class="MJX-TeXAtom-ORD">
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</mrow>
</mrow>
<mo>)</mo>
</mrow>
</mstyle>
</mrow>
{\displaystyle E_{k}={\frac {1}{2}}\sum _{i}{\frac {|{\bf {{p}_{i}|^{2}}}}{2m_{i}}}=\sum _{i}\left({\frac {{{p_{r}}_{i}}^{2}}{2m_{i}}}+{\frac {1}{2}}{\bf {{L}_{i}}}^{\textsf {T}}{I_{i}}^{-1}{\bf {{L}_{i}}}\right)}
</semantics>
</math></span><img alt="{\displaystyle E_{k}={\frac {1}{2}}\sum _{i}{\frac {|{\bf {{p}_{i}|^{2}}}}{2m_{i}}}=\sum _{i}\left({\frac {{{p_{r}}_{i}}^{2}}{2m_{i}}}+{\frac {1}{2}}{\bf {{L}_{i}}}^{\textsf {T}}{I_{i}}^{-1}{\bf {{L}_{i}}}\right)}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4add86884166e377f24c3fec726f903bdb4b00cf" style="vertical-align: -3.171ex; width:48.112ex; height:7.509ex;"/></span></dd></dl><p>where <i>p</i><sub>r</sub> is the momentum in the radial direction, and the moment of inertia is a 3-dimensional matrix; bold letters stand for 3-dimensional vectors.
</p><p>For point-like bodies we have:
</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle E_{k}=\sum _{i}\left({\frac {{{p_{r}}_{i}}^{2}}{2m_{i}}}+{\frac {|{\bf {{L}_{i}}}|^{2}}{2m_{i}{r_{i}}^{2}}}\right)}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
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<msub>
<mi>E</mi>
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<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
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<mi>i</mi>
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<mn>2</mn>
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<msub>
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<mo>+</mo>
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<mfrac>
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<mo stretchy="false">|</mo>
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<mo stretchy="false">|</mo>
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<mn>2</mn>
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<mn>2</mn>
<msub>
<mi>m</mi>
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<mi>i</mi>
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</msub>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>r</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
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</msub>
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<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
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</msup>
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</mfrac>
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<mo>)</mo>
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</mstyle>
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{\displaystyle E_{k}=\sum _{i}\left({\frac {{{p_{r}}_{i}}^{2}}{2m_{i}}}+{\frac {|{\bf {{L}_{i}}}|^{2}}{2m_{i}{r_{i}}^{2}}}\right)}
</semantics>
</math></span><img alt="{\displaystyle E_{k}=\sum _{i}\left({\frac {{{p_{r}}_{i}}^{2}}{2m_{i}}}+{\frac {|{\bf {{L}_{i}}}|^{2}}{2m_{i}{r_{i}}^{2}}}\right)}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c0aa14d194c9fb2f1e19f46e7c5f2f7a257d12a" style="vertical-align: -3.171ex; width:28.746ex; height:7.509ex;"/></span></dd></dl><p>This form of the kinetic energy part of the Hamiltonian is useful in analyzing central potential problems, and is easily transformed to a quantum mechanical work frame (e.g. in the hydrogen atom problem).
</p>samsofihttp://www.blogger.com/profile/12794342593186572847noreply@blogger.com0tag:blogger.com,1999:blog-4126202964944401087.post-66354050171599396632021-01-20T04:07:00.005-08:002021-01-20T04:07:12.068-08:00Angular momentum in orbital mechanics<img alt="" class="thumbimage" data-file-height="682" data-file-width="698" decoding="async" height="215" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/24/Ang_mom_vector_diagram.png/220px-Ang_mom_vector_diagram.png" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/24/Ang_mom_vector_diagram.png/330px-Ang_mom_vector_diagram.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/24/Ang_mom_vector_diagram.png/440px-Ang_mom_vector_diagram.png 2x" width="220"/><br/><br/><br/><p>While in classical mechanics the language of angular momentum can be replaced by Newton's laws of motion, it is particularly useful for motion in central potential such as planetary motion in the solar system. Thus, the orbit of a planet in the solar system is defined by its energy, angular momentum and angles of the orbit major axis relative to a coordinate frame.
</p><p>In astrodynamics and celestial mechanics, a <i>massless</i> (or <i>per unit mass</i>) angular momentum is defined
</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle \mathbf {h} =\mathbf {r} \times \mathbf {v} ,}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">h</mi>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">r</mi>
</mrow>
<mo>×<!-- × --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">v</mi>
</mrow>
<mo>,</mo>
</mstyle>
</mrow>
{\displaystyle \mathbf {h} =\mathbf {r} \times \mathbf {v} ,}
</semantics>
</math></span><img alt="\mathbf {h} =\mathbf {r} \times \mathbf {v} ," aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d21943fd68a6ac77006c0d7b71cedd7b3639b10" style="vertical-align: -0.671ex; width:10.584ex; height:2.509ex;"/></span></dd></dl><p>called <i>specific angular momentum</i>. Note that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle \mathbf {L} =m\mathbf {h} .}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">L</mi>
</mrow>
<mo>=</mo>
<mi>m</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">h</mi>
</mrow>
<mo>.</mo>
</mstyle>
</mrow>
{\displaystyle \mathbf {L} =m\mathbf {h} .}
</semantics>
</math></span><img alt="\mathbf {L} =m\mathbf {h} ." aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/451a7af5aa41456418306ab6657c780e020a4855" style="vertical-align: -0.338ex; width:8.879ex; height:2.176ex;"/></span> Mass is often unimportant in orbital mechanics calculations, because motion is defined by gravity. The primary body of the system is often so much larger than any bodies in motion about it that the smaller bodies have a negligible gravitational effect on it; it is, in effect, stationary. All bodies are apparently attracted by its gravity in the same way, regardless of mass, and therefore all move approximately the same way under the same conditions.
</p>samsofihttp://www.blogger.com/profile/12794342593186572847noreply@blogger.com0tag:blogger.com,1999:blog-4126202964944401087.post-8639946504644521752021-01-20T04:07:00.003-08:002021-01-20T04:07:08.441-08:00Solid bodies<img alt="" class="thumbimage" data-file-height="682" data-file-width="698" decoding="async" height="215" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/24/Ang_mom_vector_diagram.png/220px-Ang_mom_vector_diagram.png" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/24/Ang_mom_vector_diagram.png/330px-Ang_mom_vector_diagram.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/24/Ang_mom_vector_diagram.png/440px-Ang_mom_vector_diagram.png 2x" width="220"/><br/><br/><br/><p>Angular momentum is also an extremely useful concept for describing rotating rigid bodies such as a gyroscope or a rocky planet.
For a continuous mass distribution with density function <i>ρ</i>(<b>r</b>), a differential volume element <i>dV</i> with position vector <b>r</b> within the mass has a mass element <i>dm</i> = <i>ρ</i>(<b>r</b>)<i>dV</i>. Therefore, the infinitesimal angular momentum of this element is:
</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle d\mathbf {L} =\mathbf {r} \times dm\mathbf {v} =\mathbf {r} \times \rho (\mathbf {r} )dV\mathbf {v} =dV\mathbf {r} \times \rho (\mathbf {r} )\mathbf {v} }" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>d</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">L</mi>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">r</mi>
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<mo>×<!-- × --></mo>
<mi>d</mi>
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<mi mathvariant="bold">v</mi>
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<mo>=</mo>
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<mo>×<!-- × --></mo>
<mi>ρ<!-- ρ --></mi>
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<mi>d</mi>
<mi>V</mi>
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<mi mathvariant="bold">v</mi>
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<mo>=</mo>
<mi>d</mi>
<mi>V</mi>
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<mi mathvariant="bold">r</mi>
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<mo>×<!-- × --></mo>
<mi>ρ<!-- ρ --></mi>
<mo stretchy="false">(</mo>
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{\displaystyle d\mathbf {L} =\mathbf {r} \times dm\mathbf {v} =\mathbf {r} \times \rho (\mathbf {r} )dV\mathbf {v} =dV\mathbf {r} \times \rho (\mathbf {r} )\mathbf {v} }
</semantics>
</math></span><img alt="d\mathbf {L} =\mathbf {r} \times dm\mathbf {v} =\mathbf {r} \times \rho (\mathbf {r} )dV\mathbf {v} =dV\mathbf {r} \times \rho (\mathbf {r} )\mathbf {v} " aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/745bcb263750bb4b6df11130e916204d27d891f0" style="vertical-align: -0.838ex; width:45.669ex; height:2.843ex;"/></span></dd></dl><p>and integrating this differential over the volume of the entire mass gives its total angular momentum:
</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle \mathbf {L} =\int _{V}dV\mathbf {r} \times \rho (\mathbf {r} )\mathbf {v} }" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">L</mi>
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<mo>=</mo>
<msub>
<mo>∫<!-- ∫ --></mo>
<mrow class="MJX-TeXAtom-ORD">
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</msub>
<mi>d</mi>
<mi>V</mi>
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<mi mathvariant="bold">r</mi>
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<mo>×<!-- × --></mo>
<mi>ρ<!-- ρ --></mi>
<mo stretchy="false">(</mo>
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{\displaystyle \mathbf {L} =\int _{V}dV\mathbf {r} \times \rho (\mathbf {r} )\mathbf {v} }
</semantics>
</math></span><img alt="\mathbf {L} =\int _{V}dV\mathbf {r} \times \rho (\mathbf {r} )\mathbf {v} " aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0dfcf143a296d05052ee9f5612c8c78977806eea" style="vertical-align: -2.338ex; width:20.352ex; height:5.676ex;"/></span></dd></dl><p>In the derivation which follows, integrals similar to this can replace the sums for the case of continuous mass.
</p><h3><span class="mw-headline" id="Collection_of_particles">Collection of particles</span><span class="mw-editsection"><span class="mw-editsection-bracket"></span>edit<span class="mw-editsection-bracket"></span></span></h3><p>For a collection of particles in motion about an arbitrary origin, it is informative to develop the equation of angular momentum by resolving their motion into components about their own center of mass and about the origin. Given,
</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle m_{i}}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>m</mi>
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<mi>i</mi>
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{\displaystyle m_{i}}
</semantics>
</math></span><img alt="m_{i}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95ec8e804f69706d3f5ad235f4f983220c8df7c2" style="vertical-align: -0.671ex; width:2.84ex; height:2.009ex;"/></span> is the mass of particle <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle i}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
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<mi>i</mi>
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{\displaystyle i}
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</math></span><img alt="i" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;"/></span>,</dd>
<dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle \mathbf {R} _{i}}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">R</mi>
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<mi>i</mi>
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{\displaystyle \mathbf {R} _{i}}
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</math></span><img alt="\mathbf {R} _{i}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f89b7e6f1eb0602ce2df26f016b4a4a9d55f86f" style="vertical-align: -0.671ex; width:2.803ex; height:2.509ex;"/></span> is the position vector of particle <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle i}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>i</mi>
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{\displaystyle i}
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</math></span><img alt="i" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;"/></span> vs the origin,</dd>
<dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle \mathbf {V} _{i}}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
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<mi mathvariant="bold">V</mi>
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{\displaystyle \mathbf {V} _{i}}
</semantics>
</math></span><img alt="\mathbf {V} _{i}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6961baf3efe47cd95e04a2e32f917e33fe787ef3" style="vertical-align: -0.671ex; width:2.819ex; height:2.509ex;"/></span> is the velocity of particle <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle i}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>i</mi>
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{\displaystyle i}
</semantics>
</math></span><img alt="i" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;"/></span> vs the origin,</dd>
<dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle \mathbf {R} }" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">R</mi>
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{\displaystyle \mathbf {R} }
</semantics>
</math></span><img alt="\mathbf {R} " aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5de85fcc2a00d8ba14aae84aeef812d7fef4b3d5" style="vertical-align: -0.338ex; width:2.003ex; height:2.176ex;"/></span> is the position vector of the center of mass vs the origin,</dd>
<dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle \mathbf {V} }" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">V</mi>
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{\displaystyle \mathbf {V} }
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</math></span><img alt="\mathbf {V} " aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0048514530d0c0fb8a7beb795110815a818784d" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;"/></span> is the velocity of the center of mass vs the origin,</dd>
<dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle \mathbf {r} _{i}}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
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<mi mathvariant="bold">r</mi>
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<mi>i</mi>
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{\displaystyle \mathbf {r} _{i}}
</semantics>
</math></span><img alt="\mathbf {r} _{i}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed603561819ebd007acd75a0931d3ba401ad677a" style="vertical-align: -0.671ex; width:1.902ex; height:2.009ex;"/></span> is the position vector of particle <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle i}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>i</mi>
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{\displaystyle i}
</semantics>
</math></span><img alt="i" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;"/></span> vs the center of mass,</dd>
<dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle \mathbf {v} _{i}}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">v</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mstyle>
</mrow>
{\displaystyle \mathbf {v} _{i}}
</semantics>
</math></span><img alt="\mathbf {v} _{i}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51747274b58895dd357bb270ba1b5cb71e4fa355" style="vertical-align: -0.671ex; width:2.211ex; height:2.009ex;"/></span> is the velocity of particle <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle i}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>i</mi>
</mstyle>
</mrow>
{\displaystyle i}
</semantics>
</math></span><img alt="i" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;"/></span> vs the center of mass,</dd></dl><p>The total mass of the particles is simply their sum,
</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle M=\sum _{i}m_{i}.}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>M</mi>
<mo>=</mo>
<munder>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</munder>
<msub>
<mi>m</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>.</mo>
</mstyle>
</mrow>
{\displaystyle M=\sum _{i}m_{i}.}
</semantics>
</math></span><img alt="M=\sum _{i}m_{i}." aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c71e78a6457b75b0af823f3989eabc616b624b6" style="vertical-align: -3.005ex; width:12.77ex; height:5.509ex;"/></span></dd></dl><p>The position vector of the center of mass is defined by,
</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle M\mathbf {R} =\sum _{i}m_{i}\mathbf {R} _{i}.}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>M</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">R</mi>
</mrow>
<mo>=</mo>
<munder>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</munder>
<msub>
<mi>m</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">R</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>.</mo>
</mstyle>
</mrow>
{\displaystyle M\mathbf {R} =\sum _{i}m_{i}\mathbf {R} _{i}.}
</semantics>
</math></span><img alt="M\mathbf {R} =\sum _{i}m_{i}\mathbf {R} _{i}." aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0f133d75bb6cbe0958d66acc95e5e186655942e" style="vertical-align: -3.005ex; width:17.576ex; height:5.509ex;"/></span></dd></dl><p>By inspection,
</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle \mathbf {R} _{i}=\mathbf {R} +\mathbf {r} _{i}}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">R</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">R</mi>
</mrow>
<mo>+</mo>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">r</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mstyle>
</mrow>
{\displaystyle \mathbf {R} _{i}=\mathbf {R} +\mathbf {r} _{i}}
</semantics>
</math></span><img alt="\mathbf {R} _{i}=\mathbf {R} +\mathbf {r} _{i}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6002070fa6d4ad65b2b6623f4f57918222bf11ed" style="vertical-align: -0.671ex; width:12.647ex; height:2.509ex;"/></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle \mathbf {V} _{i}=\mathbf {V} +\mathbf {v} _{i}.}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">V</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">V</mi>
</mrow>
<mo>+</mo>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">v</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>.</mo>
</mstyle>
</mrow>
{\displaystyle \mathbf {V} _{i}=\mathbf {V} +\mathbf {v} _{i}.}
</semantics>
</math></span><img alt="\mathbf {V} _{i}=\mathbf {V} +\mathbf {v} _{i}." aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f91b76b7f70b2baa710308cf0f00879cda12a9b" style="vertical-align: -0.671ex; width:13.635ex; height:2.509ex;"/></span></dd></dl><p>The total angular momentum of the collection of particles is the sum of the angular momentum of each particle,
</p><p>Expanding <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle \mathbf {R} _{i}}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">R</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mstyle>
</mrow>
{\displaystyle \mathbf {R} _{i}}
</semantics>
</math></span><img alt="\mathbf {R} _{i}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f89b7e6f1eb0602ce2df26f016b4a4a9d55f86f" style="vertical-align: -0.671ex; width:2.803ex; height:2.509ex;"/></span>,
</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle {\begin{aligned}\mathbf {L} &=\sum _{i}\left\left(\mathbf {R} +\mathbf {r} _{i}\right)\times m_{i}\mathbf {V} _{i}\right\\&=\sum _{i}\left\mathbf {R} \times m_{i}\mathbf {V} _{i}+\mathbf {r} _{i}\times m_{i}\mathbf {V} _{i}\right\end{aligned}}}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mtable columnalign="right left right left right left right left right left right left" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true" rowspacing="3pt">
<mtr>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">L</mi>
</mrow>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<munder>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</munder>
<mrow>
<mo></mo>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">R</mi>
</mrow>
<mo>+</mo>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">r</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mrow>
<mo>)</mo>
</mrow>
<mo>×<!-- × --></mo>
<msub>
<mi>m</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">V</mi>
</mrow>
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<mi>i</mi>
</mrow>
</msub>
</mrow>
<mo></mo>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd></mtd>
<mtd>
<mi></mi>
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<munder>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</munder>
<mrow>
<mo></mo>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">R</mi>
</mrow>
<mo>×<!-- × --></mo>
<msub>
<mi>m</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">V</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">r</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>×<!-- × --></mo>
<msub>
<mi>m</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">V</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mrow>
<mo></mo>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
</mstyle>
</mrow>
{\displaystyle {\begin{aligned}\mathbf {L} &=\sum _{i}\left\left(\mathbf {R} +\mathbf {r} _{i}\right)\times m_{i}\mathbf {V} _{i}\right\\&=\sum _{i}\left\mathbf {R} \times m_{i}\mathbf {V} _{i}+\mathbf {r} _{i}\times m_{i}\mathbf {V} _{i}\right\end{aligned}}}
</semantics>
</math></span><img alt="{\displaystyle {\begin{aligned}\mathbf {L} &=\sum _{i}\left\left(\mathbf {R} +\mathbf {r} _{i}\right)\times m_{i}\mathbf {V} _{i}\right\\&=\sum _{i}\left\mathbf {R} \times m_{i}\mathbf {V} _{i}+\mathbf {r} _{i}\times m_{i}\mathbf {V} _{i}\right\end{aligned}}}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77077ef30b49ac992b78c6a4c6446709bb1720a9" style="vertical-align: -5.005ex; width:34.239ex; height:11.176ex;"/></span></dd></dl><p>Expanding <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle \mathbf {V} _{i}}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">V</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mstyle>
</mrow>
{\displaystyle \mathbf {V} _{i}}
</semantics>
</math></span><img alt="\mathbf {V} _{i}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6961baf3efe47cd95e04a2e32f917e33fe787ef3" style="vertical-align: -0.671ex; width:2.819ex; height:2.509ex;"/></span>,
</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle {\begin{aligned}\mathbf {L} &=\sum _{i}\left\mathbf {R} \times m_{i}\left(\mathbf {V} +\mathbf {v} _{i}\right)+\mathbf {r} _{i}\times m_{i}(\mathbf {V} +\mathbf {v} _{i})\right\\&=\sum _{i}\left\mathbf {R} \times m_{i}\mathbf {V} +\mathbf {R} \times m_{i}\mathbf {v} _{i}+\mathbf {r} _{i}\times m_{i}\mathbf {V} +\mathbf {r} _{i}\times m_{i}\mathbf {v} _{i}\right\\&=\sum _{i}\mathbf {R} \times m_{i}\mathbf {V} +\sum _{i}\mathbf {R} \times m_{i}\mathbf {v} _{i}+\sum _{i}\mathbf {r} _{i}\times m_{i}\mathbf {V} +\sum _{i}\mathbf {r} _{i}\times m_{i}\mathbf {v} _{i}\end{aligned}}}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mtable columnalign="right left right left right left right left right left right left" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true" rowspacing="3pt">
<mtr>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">L</mi>
</mrow>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<munder>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</munder>
<mrow>
<mo></mo>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">R</mi>
</mrow>
<mo>×<!-- × --></mo>
<msub>
<mi>m</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mrow>
<mo>(</mo>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">V</mi>
</mrow>
<mo>+</mo>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">v</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">r</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>×<!-- × --></mo>
<msub>
<mi>m</mi>
<mrow class="MJX-TeXAtom-ORD">
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</mrow>
</msub>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">V</mi>
</mrow>
<mo>+</mo>
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<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">v</mi>
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<mrow class="MJX-TeXAtom-ORD">
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</mrow>
</msub>
<mo stretchy="false">)</mo>
</mrow>
<mo></mo>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd></mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<munder>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
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</mrow>
</munder>
<mrow>
<mo></mo>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
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</mrow>
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<msub>
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<mrow class="MJX-TeXAtom-ORD">
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</mrow>
</msub>
<mrow class="MJX-TeXAtom-ORD">
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<msub>
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<mrow class="MJX-TeXAtom-ORD">
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<mi mathvariant="bold">v</mi>
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</mrow>
<mo></mo>
</mrow>
</mtd>
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</msub>
<mrow class="MJX-TeXAtom-ORD">
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</mrow>
<mo>+</mo>
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<mo>∑<!-- ∑ --></mo>
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</mrow>
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</mrow>
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<msub>
<mi>m</mi>
<mrow class="MJX-TeXAtom-ORD">
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{\displaystyle {\begin{aligned}\mathbf {L} &=\sum _{i}\left\mathbf {R} \times m_{i}\left(\mathbf {V} +\mathbf {v} _{i}\right)+\mathbf {r} _{i}\times m_{i}(\mathbf {V} +\mathbf {v} _{i})\right\\&=\sum _{i}\left\mathbf {R} \times m_{i}\mathbf {V} +\mathbf {R} \times m_{i}\mathbf {v} _{i}+\mathbf {r} _{i}\times m_{i}\mathbf {V} +\mathbf {r} _{i}\times m_{i}\mathbf {v} _{i}\right\\&=\sum _{i}\mathbf {R} \times m_{i}\mathbf {V} +\sum _{i}\mathbf {R} \times m_{i}\mathbf {v} _{i}+\sum _{i}\mathbf {r} _{i}\times m_{i}\mathbf {V} +\sum _{i}\mathbf {r} _{i}\times m_{i}\mathbf {v} _{i}\end{aligned}}}
</semantics>
</math></span><img alt="{\displaystyle {\begin{aligned}\mathbf {L} &=\sum _{i}\left\mathbf {R} \times m_{i}\left(\mathbf {V} +\mathbf {v} _{i}\right)+\mathbf {r} _{i}\times m_{i}(\mathbf {V} +\mathbf {v} _{i})\right\\&=\sum _{i}\left\mathbf {R} \times m_{i}\mathbf {V} +\mathbf {R} \times m_{i}\mathbf {v} _{i}+\mathbf {r} _{i}\times m_{i}\mathbf {V} +\mathbf {r} _{i}\times m_{i}\mathbf {v} _{i}\right\\&=\sum _{i}\mathbf {R} \times m_{i}\mathbf {V} +\sum _{i}\mathbf {R} \times m_{i}\mathbf {v} _{i}+\sum _{i}\mathbf {r} _{i}\times m_{i}\mathbf {V} +\sum _{i}\mathbf {r} _{i}\times m_{i}\mathbf {v} _{i}\end{aligned}}}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4784bd78e4066783f47bd72472bfa49a0887b0bf" style="vertical-align: -7.838ex; width:67.94ex; height:16.843ex;"/></span></dd></dl><p>It can be shown that (see sidebar),
</p><table cellspacing="5" class="toccolours" style="float:right; margin-left:0.5em; margin-right:0.5em; font-size:84%; background:white; color:black; width:30em; max-width:30%;">
<tbody><tr>
<td style="text-align:center;">
<p><b>Prove that</b> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle \sum _{i}m_{i}\mathbf {r} _{i}=\mathbf {0} }" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
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{\displaystyle \sum _{i}m_{i}\mathbf {r} _{i}=\mathbf {0} }
</semantics>
</math></span><img alt="\sum _{i}m_{i}\mathbf {r} _{i}=\mathbf {0} " aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9534099a1221dce32185b2cd617fe516c15618f8" style="vertical-align: -3.005ex; width:12.919ex; height:5.509ex;"/></span>
</p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle {\begin{aligned}\mathbf {r} _{i}&=\mathbf {R} _{i}-\mathbf {R} \\m_{i}\mathbf {r} _{i}&=m_{i}\left(\mathbf {R} _{i}-\mathbf {R} \right)\\\sum _{i}m_{i}\mathbf {r} _{i}&=\sum _{i}m_{i}\left(\mathbf {R} _{i}-\mathbf {R} \right)\\&=\sum _{i}(m_{i}\mathbf {R} _{i}-m_{i}\mathbf {R} )\\&=\sum _{i}m_{i}\mathbf {R} _{i}-\sum _{i}m_{i}\mathbf {R} \\&=\sum _{i}m_{i}\mathbf {R} _{i}-\left(\sum _{i}m_{i}\right)\mathbf {R} \\&=\sum _{i}m_{i}\mathbf {R} _{i}-M\mathbf {R} \end{aligned}}}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
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{\displaystyle {\begin{aligned}\mathbf {r} _{i}&=\mathbf {R} _{i}-\mathbf {R} \\m_{i}\mathbf {r} _{i}&=m_{i}\left(\mathbf {R} _{i}-\mathbf {R} \right)\\\sum _{i}m_{i}\mathbf {r} _{i}&=\sum _{i}m_{i}\left(\mathbf {R} _{i}-\mathbf {R} \right)\\&=\sum _{i}(m_{i}\mathbf {R} _{i}-m_{i}\mathbf {R} )\\&=\sum _{i}m_{i}\mathbf {R} _{i}-\sum _{i}m_{i}\mathbf {R} \\&=\sum _{i}m_{i}\mathbf {R} _{i}-\left(\sum _{i}m_{i}\right)\mathbf {R} \\&=\sum _{i}m_{i}\mathbf {R} _{i}-M\mathbf {R} \end{aligned}}}
</semantics>
</math></span><img alt="{\displaystyle {\begin{aligned}\mathbf {r} _{i}&=\mathbf {R} _{i}-\mathbf {R} \\m_{i}\mathbf {r} _{i}&=m_{i}\left(\mathbf {R} _{i}-\mathbf {R} \right)\\\sum _{i}m_{i}\mathbf {r} _{i}&=\sum _{i}m_{i}\left(\mathbf {R} _{i}-\mathbf {R} \right)\\&=\sum _{i}(m_{i}\mathbf {R} _{i}-m_{i}\mathbf {R} )\\&=\sum _{i}m_{i}\mathbf {R} _{i}-\sum _{i}m_{i}\mathbf {R} \\&=\sum _{i}m_{i}\mathbf {R} _{i}-\left(\sum _{i}m_{i}\right)\mathbf {R} \\&=\sum _{i}m_{i}\mathbf {R} _{i}-M\mathbf {R} \end{aligned}}}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc97849976c304bb00a4ad72d312b17ee8f00a93" style="vertical-align: -17.343ex; margin-bottom: -0.329ex; width:37.213ex; height:36.509ex;"/></span>
</p><p>which, by the definition of the center of mass, is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle \mathbf {0} ,}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
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{\displaystyle \mathbf {0} ,}
</semantics>
</math></span><img alt="{\displaystyle \mathbf {0} ,}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2118daab5078173da3eebca1673a98c25d213869" style="vertical-align: -0.671ex; width:1.983ex; height:2.509ex;"/></span> and similarly for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle \sum _{i}m_{i}\mathbf {v} _{i}.}" xmlns="http://www.w3.org/1998/Math/MathML">
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{\displaystyle \sum _{i}m_{i}\mathbf {v} _{i}.}
</semantics>
</math></span><img alt="\sum _{i}m_{i}\mathbf {v} _{i}." aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5adb9c53dafaa6ebce889bf74345fb066e75ed4e" style="vertical-align: -3.005ex; width:9.44ex; height:5.509ex;"/></span>
</p>
</td></tr></tbody></table><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle \sum _{i}m_{i}\mathbf {r} _{i}=\mathbf {0} }" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<munder>
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{\displaystyle \sum _{i}m_{i}\mathbf {r} _{i}=\mathbf {0} }
</semantics>
</math></span><img alt="\sum _{i}m_{i}\mathbf {r} _{i}=\mathbf {0} " aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9534099a1221dce32185b2cd617fe516c15618f8" style="vertical-align: -3.005ex; width:12.919ex; height:5.509ex;"/></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle \sum _{i}m_{i}\mathbf {v} _{i}=\mathbf {0} ,}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
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{\displaystyle \sum _{i}m_{i}\mathbf {v} _{i}=\mathbf {0} ,}
</semantics>
</math></span><img alt="{\displaystyle \sum _{i}m_{i}\mathbf {v} _{i}=\mathbf {0} ,}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd78a68dda7e71ed85c4882689859209270d92a1" style="vertical-align: -3.005ex; width:13.875ex; height:5.509ex;"/></span></dd></dl><p>therefore the second and third terms vanish,
</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle \mathbf {L} =\sum _{i}\mathbf {R} \times m_{i}\mathbf {V} +\sum _{i}\mathbf {r} _{i}\times m_{i}\mathbf {v} _{i}.}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
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{\displaystyle \mathbf {L} =\sum _{i}\mathbf {R} \times m_{i}\mathbf {V} +\sum _{i}\mathbf {r} _{i}\times m_{i}\mathbf {v} _{i}.}
</semantics>
</math></span><img alt="\mathbf {L} =\sum _{i}\mathbf {R} \times m_{i}\mathbf {V} +\sum _{i}\mathbf {r} _{i}\times m_{i}\mathbf {v} _{i}." aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/adda39d405afd75d159a23c9a06a46a21bf8f221" style="vertical-align: -3.005ex; width:35.174ex; height:5.509ex;"/></span></dd></dl><p>The first term can be rearranged,
</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle \sum _{i}\mathbf {R} \times m_{i}\mathbf {V} =\mathbf {R} \times \sum _{i}m_{i}\mathbf {V} =\mathbf {R} \times M\mathbf {V} ,}" xmlns="http://www.w3.org/1998/Math/MathML">
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<msub>
<mi>m</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">V</mi>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">R</mi>
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<mo>×<!-- × --></mo>
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<mo>∑<!-- ∑ --></mo>
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<mi>i</mi>
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<mi>m</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
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</msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">V</mi>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">R</mi>
</mrow>
<mo>×<!-- × --></mo>
<mi>M</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">V</mi>
</mrow>
<mo>,</mo>
</mstyle>
</mrow>
{\displaystyle \sum _{i}\mathbf {R} \times m_{i}\mathbf {V} =\mathbf {R} \times \sum _{i}m_{i}\mathbf {V} =\mathbf {R} \times M\mathbf {V} ,}
</semantics>
</math></span><img alt="\sum _{i}\mathbf {R} \times m_{i}\mathbf {V} =\mathbf {R} \times \sum _{i}m_{i}\mathbf {V} =\mathbf {R} \times M\mathbf {V} ," aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4100b708f2401085dd24879939ba51656f69e8d8" style="vertical-align: -3.005ex; width:43.04ex; height:5.509ex;"/></span></dd></dl><p>and total angular momentum for the collection of particles is finally,
</p><p>The first term is the angular momentum of the center of mass relative to the origin. Similar to <b>Single particle</b>, below, it is the angular momentum of one particle of mass <i>M</i> at the center of mass moving with velocity <b>V</b>. The second term is the angular momentum of the particles moving relative to the center of mass, similar to <b>Fixed center of mass</b>, below. The result is general—the motion of the particles is not restricted to rotation or revolution about the origin or center of mass. The particles need not be individual masses, but can be elements of a continuous distribution, such as a solid body.
</p><p>Rearranging equation (<b>2</b>) by vector identities, multiplying both terms by "one", and grouping appropriately,
</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle {\begin{aligned}\mathbf {L} &=M(\mathbf {R} \times \mathbf {V} )+\sum _{i}\leftm_{i}\left(\mathbf {r} _{i}\times \mathbf {v} _{i}\right)\right,\\&={\frac {R^{2}}{R^{2}}}M\left(\mathbf {R} \times \mathbf {V} \right)+\sum _{i}\left{\frac {r_{i}^{2}}{r_{i}^{2}}}m_{i}\left(\mathbf {r} _{i}\times \mathbf {v} _{i}\right)\right,\\&=R^{2}M\left({\frac {\mathbf {R} \times \mathbf {V} }{R^{2}}}\right)+\sum _{i}\leftr_{i}^{2}m_{i}\left({\frac {\mathbf {r} _{i}\times \mathbf {v} _{i}}{r_{i}^{2}}}\right)\right,\\\end{aligned}}}" xmlns="http://www.w3.org/1998/Math/MathML">
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<mi mathvariant="bold">L</mi>
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<mtd>
<mi></mi>
<mo>=</mo>
<mi>M</mi>
<mo stretchy="false">(</mo>
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<mi mathvariant="bold">R</mi>
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<mo>×<!-- × --></mo>
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<mi mathvariant="bold">V</mi>
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<mo stretchy="false">)</mo>
<mo>+</mo>
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<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
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<mo></mo>
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<msub>
<mi>m</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mrow>
<mo>(</mo>
<mrow>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">r</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
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</msub>
<mo>×<!-- × --></mo>
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<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">v</mi>
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<mi>i</mi>
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<mo>)</mo>
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<mo></mo>
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<mo>,</mo>
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<mtd></mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<msup>
<mi>R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
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</msup>
<msup>
<mi>R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
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</msup>
</mfrac>
</mrow>
<mi>M</mi>
<mrow>
<mo>(</mo>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">R</mi>
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<mo>×<!-- × --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">V</mi>
</mrow>
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<mo>)</mo>
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<mo>+</mo>
<munder>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
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<mo></mo>
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<mfrac>
<msubsup>
<mi>r</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
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<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
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</msubsup>
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<mi>r</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
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<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
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</mfrac>
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<msub>
<mi>m</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
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</msub>
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<mo>(</mo>
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<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">r</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
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</msub>
<mo>×<!-- × --></mo>
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<mi mathvariant="bold">v</mi>
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<mi>i</mi>
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</msub>
</mrow>
<mo>)</mo>
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<mo></mo>
</mrow>
<mo>,</mo>
</mtd>
</mtr>
<mtr>
<mtd></mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<msup>
<mi>R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mi>M</mi>
<mrow>
<mo>(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">R</mi>
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<mo>×<!-- × --></mo>
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<mi mathvariant="bold">V</mi>
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<mi>R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
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<mo>)</mo>
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<mo>+</mo>
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<mo>∑<!-- ∑ --></mo>
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<mi>i</mi>
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<mi>r</mi>
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<mn>2</mn>
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<mi>m</mi>
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<mi>i</mi>
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<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">r</mi>
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<mi>i</mi>
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<mo>×<!-- × --></mo>
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<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">v</mi>
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<mi>r</mi>
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<mi>i</mi>
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<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
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<mo>)</mo>
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<mo></mo>
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<mo>,</mo>
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{\displaystyle {\begin{aligned}\mathbf {L} &=M(\mathbf {R} \times \mathbf {V} )+\sum _{i}\leftm_{i}\left(\mathbf {r} _{i}\times \mathbf {v} _{i}\right)\right,\\&={\frac {R^{2}}{R^{2}}}M\left(\mathbf {R} \times \mathbf {V} \right)+\sum _{i}\left{\frac {r_{i}^{2}}{r_{i}^{2}}}m_{i}\left(\mathbf {r} _{i}\times \mathbf {v} _{i}\right)\right,\\&=R^{2}M\left({\frac {\mathbf {R} \times \mathbf {V} }{R^{2}}}\right)+\sum _{i}\leftr_{i}^{2}m_{i}\left({\frac {\mathbf {r} _{i}\times \mathbf {v} _{i}}{r_{i}^{2}}}\right)\right,\\\end{aligned}}}
</semantics>
</math></span><img alt="{\displaystyle {\begin{aligned}\mathbf {L} &=M(\mathbf {R} \times \mathbf {V} )+\sum _{i}\leftm_{i}\left(\mathbf {r} _{i}\times \mathbf {v} _{i}\right)\right,\\&={\frac {R^{2}}{R^{2}}}M\left(\mathbf {R} \times \mathbf {V} \right)+\sum _{i}\left{\frac {r_{i}^{2}}{r_{i}^{2}}}m_{i}\left(\mathbf {r} _{i}\times \mathbf {v} _{i}\right)\right,\\&=R^{2}M\left({\frac {\mathbf {R} \times \mathbf {V} }{R^{2}}}\right)+\sum _{i}\leftr_{i}^{2}m_{i}\left({\frac {\mathbf {r} _{i}\times \mathbf {v} _{i}}{r_{i}^{2}}}\right)\right,\\\end{aligned}}}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5324c7fa7e5e94893f678fab046c2538416bc51" style="vertical-align: -9.579ex; margin-bottom: -0.259ex; width:49.353ex; height:20.843ex;"/></span></dd></dl><p>gives the total angular momentum of the system of particles in terms of moment of inertia <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle I}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>I</mi>
</mstyle>
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{\displaystyle I}
</semantics>
</math></span><img alt="I" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;"/></span> and angular velocity <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle {\boldsymbol {\omega }}}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold-italic">ω<!-- ω --></mi>
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{\displaystyle {\boldsymbol {\omega }}}
</semantics>
</math></span><img alt="{\boldsymbol {\omega }}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7cb8af7a2f64af348e559652b6b1f0d2415ba444" style="vertical-align: -0.338ex; width:1.669ex; height:1.676ex;"/></span>,
</p><h4><span class="mw-headline" id="Single_particle_case">Single particle case</span><span class="mw-editsection"><span class="mw-editsection-bracket"></span>edit<span class="mw-editsection-bracket"></span></span></h4><p>In the case of a single particle moving about the arbitrary origin,
</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle {\begin{aligned}\mathbf {r} _{i}&=\mathbf {v} _{i}=\mathbf {0} ,\\\mathbf {r} &=\mathbf {R} ,\\\mathbf {v} &=\mathbf {V} ,\\m&=M,\end{aligned}}}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
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<mtd>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">r</mi>
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<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
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</msub>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">v</mi>
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<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
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<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn mathvariant="bold">0</mn>
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<mo>,</mo>
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<mtr>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">r</mi>
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<mtd>
<mi></mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">R</mi>
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<mo>,</mo>
</mtd>
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<mtr>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">v</mi>
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<mtd>
<mi></mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">V</mi>
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<mo>,</mo>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>m</mi>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mi>M</mi>
<mo>,</mo>
</mtd>
</mtr>
</mtable>
</mrow>
</mstyle>
</mrow>
{\displaystyle {\begin{aligned}\mathbf {r} _{i}&=\mathbf {v} _{i}=\mathbf {0} ,\\\mathbf {r} &=\mathbf {R} ,\\\mathbf {v} &=\mathbf {V} ,\\m&=M,\end{aligned}}}
</semantics>
</math></span><img alt="{\displaystyle {\begin{aligned}\mathbf {r} _{i}&=\mathbf {v} _{i}=\mathbf {0} ,\\\mathbf {r} &=\mathbf {R} ,\\\mathbf {v} &=\mathbf {V} ,\\m&=M,\end{aligned}}}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4257d4b04ffe20195d98c78c9963ba27e3e0aa86" style="vertical-align: -5.338ex; width:13.183ex; height:11.843ex;"/></span></dd>
<dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle \sum _{i}\mathbf {r} _{i}\times m_{i}\mathbf {v} _{i}=\mathbf {0} ,}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<munder>
<mo>∑<!-- ∑ --></mo>
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</munder>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">r</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
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</msub>
<mo>×<!-- × --></mo>
<msub>
<mi>m</mi>
<mrow class="MJX-TeXAtom-ORD">
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<mrow class="MJX-TeXAtom-ORD">
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{\displaystyle \sum _{i}\mathbf {r} _{i}\times m_{i}\mathbf {v} _{i}=\mathbf {0} ,}
</semantics>
</math></span><img alt="\sum _{i}\mathbf {r} _{i}\times m_{i}\mathbf {v} _{i}=\mathbf {0} ," aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/944a683c88b75f2a6c1988793c7f82ea7fb0b95f" style="vertical-align: -3.005ex; width:18.617ex; height:5.509ex;"/></span></dd>
<dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle \sum _{i}I_{i}{\boldsymbol {\omega }}_{i}=\mathbf {0} ,}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<munder>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</munder>
<msub>
<mi>I</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold-italic">ω<!-- ω --></mi>
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<mn mathvariant="bold">0</mn>
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<mo>,</mo>
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{\displaystyle \sum _{i}I_{i}{\boldsymbol {\omega }}_{i}=\mathbf {0} ,}
</semantics>
</math></span><img alt="\sum _{i}I_{i}{\boldsymbol {\omega }}_{i}=\mathbf {0} ," aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ba328add1d5c9eea7ae394ed9c8ec4bb65d1214" style="vertical-align: -3.005ex; width:13.115ex; height:5.509ex;"/></span> and equations (<b>2</b>) and (<b>3</b>) for total angular momentum reduce to,</dd>
<dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle \mathbf {L} =\mathbf {R} \times m\mathbf {V} =I_{R}{\boldsymbol {\omega }}_{R}.}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">L</mi>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">R</mi>
</mrow>
<mo>×<!-- × --></mo>
<mi>m</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">V</mi>
</mrow>
<mo>=</mo>
<msub>
<mi>I</mi>
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<mi>R</mi>
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</msub>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold-italic">ω<!-- ω --></mi>
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<mrow class="MJX-TeXAtom-ORD">
<mi>R</mi>
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</msub>
<mo>.</mo>
</mstyle>
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{\displaystyle \mathbf {L} =\mathbf {R} \times m\mathbf {V} =I_{R}{\boldsymbol {\omega }}_{R}.}
</semantics>
</math></span><img alt="\mathbf {L} =\mathbf {R} \times m\mathbf {V} =I_{R}{\boldsymbol {\omega }}_{R}." aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71c808752c6f3802a2598be401f3a190232f4763" style="vertical-align: -0.671ex; width:23.007ex; height:2.509ex;"/></span></dd></dl><h4><span class="mw-headline" id="Case_of_a_fixed_center_of_mass">Case of a fixed center of mass</span><span class="mw-editsection"><span class="mw-editsection-bracket"></span>edit<span class="mw-editsection-bracket"></span></span></h4><p>For the case of the center of mass fixed in space with respect to the origin,
</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle \mathbf {V} =\mathbf {0} ,}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">V</mi>
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<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn mathvariant="bold">0</mn>
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<mo>,</mo>
</mstyle>
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{\displaystyle \mathbf {V} =\mathbf {0} ,}
</semantics>
</math></span><img alt="\mathbf {V} =\mathbf {0} ," aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8beda55f95c66718b581190ccc13aae081d2fd5b" style="vertical-align: -0.671ex; width:7.101ex; height:2.509ex;"/></span></dd>
<dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle \mathbf {R} \times M\mathbf {V} =\mathbf {0} ,}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">R</mi>
</mrow>
<mo>×<!-- × --></mo>
<mi>M</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">V</mi>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn mathvariant="bold">0</mn>
</mrow>
<mo>,</mo>
</mstyle>
</mrow>
{\displaystyle \mathbf {R} \times M\mathbf {V} =\mathbf {0} ,}
</semantics>
</math></span><img alt="\mathbf {R} \times M\mathbf {V} =\mathbf {0} ," aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce3333259064e5dfb948d851f5b896b346d43eff" style="vertical-align: -0.671ex; width:14.387ex; height:2.509ex;"/></span></dd>
<dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle I_{R}{\boldsymbol {\omega }}_{R}=\mathbf {0} ,}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>I</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>R</mi>
</mrow>
</msub>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold-italic">ω<!-- ω --></mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>R</mi>
</mrow>
</msub>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn mathvariant="bold">0</mn>
</mrow>
<mo>,</mo>
</mstyle>
</mrow>
{\displaystyle I_{R}{\boldsymbol {\omega }}_{R}=\mathbf {0} ,}
</semantics>
</math></span><img alt="I_{R}{\boldsymbol {\omega }}_{R}=\mathbf {0} ," aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1e57764d122bf124ac896dda2e31001565982a1" style="vertical-align: -0.671ex; width:10.733ex; height:2.509ex;"/></span> and equations (<b>2</b>) and (<b>3</b>) for total angular momentum reduce to,</dd>
<dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle \mathbf {L} =\sum _{i}\mathbf {r} _{i}\times m_{i}\mathbf {v} _{i}=\sum _{i}I_{i}{\boldsymbol {\omega }}_{i}.}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">L</mi>
</mrow>
<mo>=</mo>
<munder>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</munder>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">r</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>×<!-- × --></mo>
<msub>
<mi>m</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">v</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>=</mo>
<munder>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</munder>
<msub>
<mi>I</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold-italic">ω<!-- ω --></mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>.</mo>
</mstyle>
</mrow>
{\displaystyle \mathbf {L} =\sum _{i}\mathbf {r} _{i}\times m_{i}\mathbf {v} _{i}=\sum _{i}I_{i}{\boldsymbol {\omega }}_{i}.}
</semantics>
</math></span><img alt="\mathbf {L} =\sum _{i}\mathbf {r} _{i}\times m_{i}\mathbf {v} _{i}=\sum _{i}I_{i}{\boldsymbol {\omega }}_{i}." aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf008451f7642e1b64ec6c32d2f5a1b538c9ac01" style="vertical-align: -3.005ex; width:30.02ex; height:5.509ex;"/></span></dd></dl>samsofihttp://www.blogger.com/profile/12794342593186572847noreply@blogger.com0tag:blogger.com,1999:blog-4126202964944401087.post-80171152652809902112021-01-20T04:07:00.001-08:002021-01-20T04:07:04.076-08:00Angular momentum in general relativity<img alt="" class="thumbimage" data-file-height="239" data-file-width="350" decoding="async" height="150" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7c/Angular_momentum_bivector_and_pseudovector.svg/220px-Angular_momentum_bivector_and_pseudovector.svg.png" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7c/Angular_momentum_bivector_and_pseudovector.svg/330px-Angular_momentum_bivector_and_pseudovector.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7c/Angular_momentum_bivector_and_pseudovector.svg/440px-Angular_momentum_bivector_and_pseudovector.svg.png 2x" width="220"/><br/><br/><br/><p>In modern (20th century) theoretical physics, angular momentum (not including any intrinsic angular momentum – see below) is described using a different formalism, instead of a classical pseudovector. In this formalism, angular momentum is the 2-form Noether charge associated with rotational invariance. As a result, angular momentum is not conserved for general curved spacetimes, unless it happens to be asymptotically rotationally invariant.<i><span title="This claim needs references to reliable sources. (May 2013)">citation needed</span></i>
</p><p>In classical mechanics, the angular momentum of a particle can be reinterpreted as a plane element:
</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle \mathbf {L} =\mathbf {r} \wedge \mathbf {p} \,,}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">L</mi>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">r</mi>
</mrow>
<mo>∧<!-- ∧ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">p</mi>
</mrow>
<mspace width="thinmathspace"></mspace>
<mo>,</mo>
</mstyle>
</mrow>
{\displaystyle \mathbf {L} =\mathbf {r} \wedge \mathbf {p} \,,}
</semantics>
</math></span><img alt="{\displaystyle \mathbf {L} =\mathbf {r} \wedge \mathbf {p} \,,}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a728ebb909aecdf46e42d9d76e7ef344fa0bf0ba" style="vertical-align: -0.671ex; width:10.911ex; height:2.509ex;"/></span></dd></dl><p>in which the exterior product ∧ replaces the cross product × (these products have similar characteristics but are nonequivalent). This has the advantage of a clearer geometric interpretation as a plane element, defined from the <b>x</b> and <b>p</b> vectors, and the expression is true in any number of dimensions (two or higher). In Cartesian coordinates:
</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle {\begin{aligned}\mathbf {L} &=\left(xp_{y}-yp_{x}\right)\mathbf {e} _{x}\wedge \mathbf {e} _{y}+\left(yp_{z}-zp_{y}\right)\mathbf {e} _{y}\wedge \mathbf {e} _{z}+\left(zp_{x}-xp_{z}\right)\mathbf {e} _{z}\wedge \mathbf {e} _{x}\\&=L_{xy}\mathbf {e} _{x}\wedge \mathbf {e} _{y}+L_{yz}\mathbf {e} _{y}\wedge \mathbf {e} _{z}+L_{zx}\mathbf {e} _{z}\wedge \mathbf {e} _{x}\,,\end{aligned}}}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mtable columnalign="right left right left right left right left right left right left" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true" rowspacing="3pt">
<mtr>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">L</mi>
</mrow>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<msub>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>y</mi>
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</msub>
<mo>−<!-- − --></mo>
<mi>y</mi>
<msub>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>x</mi>
</mrow>
</msub>
</mrow>
<mo>)</mo>
</mrow>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">e</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>x</mi>
</mrow>
</msub>
<mo>∧<!-- ∧ --></mo>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">e</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>y</mi>
</mrow>
</msub>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mi>y</mi>
<msub>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>z</mi>
</mrow>
</msub>
<mo>−<!-- − --></mo>
<mi>z</mi>
<msub>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>y</mi>
</mrow>
</msub>
</mrow>
<mo>)</mo>
</mrow>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">e</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>y</mi>
</mrow>
</msub>
<mo>∧<!-- ∧ --></mo>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">e</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>z</mi>
</mrow>
</msub>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mi>z</mi>
<msub>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>x</mi>
</mrow>
</msub>
<mo>−<!-- − --></mo>
<mi>x</mi>
<msub>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>z</mi>
</mrow>
</msub>
</mrow>
<mo>)</mo>
</mrow>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">e</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>z</mi>
</mrow>
</msub>
<mo>∧<!-- ∧ --></mo>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">e</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>x</mi>
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</msub>
</mtd>
</mtr>
<mtr>
<mtd></mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<msub>
<mi>L</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>x</mi>
<mi>y</mi>
</mrow>
</msub>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">e</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>x</mi>
</mrow>
</msub>
<mo>∧<!-- ∧ --></mo>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">e</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>y</mi>
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</msub>
<mo>+</mo>
<msub>
<mi>L</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>y</mi>
<mi>z</mi>
</mrow>
</msub>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">e</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>y</mi>
</mrow>
</msub>
<mo>∧<!-- ∧ --></mo>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">e</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>z</mi>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mi>L</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>z</mi>
<mi>x</mi>
</mrow>
</msub>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">e</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>z</mi>
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</msub>
<mo>∧<!-- ∧ --></mo>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">e</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>x</mi>
</mrow>
</msub>
<mspace width="thinmathspace"></mspace>
<mo>,</mo>
</mtd>
</mtr>
</mtable>
</mrow>
</mstyle>
</mrow>
{\displaystyle {\begin{aligned}\mathbf {L} &=\left(xp_{y}-yp_{x}\right)\mathbf {e} _{x}\wedge \mathbf {e} _{y}+\left(yp_{z}-zp_{y}\right)\mathbf {e} _{y}\wedge \mathbf {e} _{z}+\left(zp_{x}-xp_{z}\right)\mathbf {e} _{z}\wedge \mathbf {e} _{x}\\&=L_{xy}\mathbf {e} _{x}\wedge \mathbf {e} _{y}+L_{yz}\mathbf {e} _{y}\wedge \mathbf {e} _{z}+L_{zx}\mathbf {e} _{z}\wedge \mathbf {e} _{x}\,,\end{aligned}}}
</semantics>
</math></span><img alt="{\displaystyle {\begin{aligned}\mathbf {L} &=\left(xp_{y}-yp_{x}\right)\mathbf {e} _{x}\wedge \mathbf {e} _{y}+\left(yp_{z}-zp_{y}\right)\mathbf {e} _{y}\wedge \mathbf {e} _{z}+\left(zp_{x}-xp_{z}\right)\mathbf {e} _{z}\wedge \mathbf {e} _{x}\\&=L_{xy}\mathbf {e} _{x}\wedge \mathbf {e} _{y}+L_{yz}\mathbf {e} _{y}\wedge \mathbf {e} _{z}+L_{zx}\mathbf {e} _{z}\wedge \mathbf {e} _{x}\,,\end{aligned}}}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/771af9385db3ca15515c2538fa62846f3fe6194e" style="vertical-align: -2.505ex; width:68.405ex; height:6.176ex;"/></span></dd></dl><p>or more compactly in index notation:
</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle L_{ij}=x_{i}p_{j}-x_{j}p_{i}\,.}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>L</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>j</mi>
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<mo>=</mo>
<msub>
<mi>x</mi>
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<msub>
<mi>p</mi>
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<mi>j</mi>
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</msub>
<mo>−<!-- − --></mo>
<msub>
<mi>x</mi>
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<mi>j</mi>
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</msub>
<msub>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mspace width="thinmathspace"></mspace>
<mo>.</mo>
</mstyle>
</mrow>
{\displaystyle L_{ij}=x_{i}p_{j}-x_{j}p_{i}\,.}
</semantics>
</math></span><img alt="{\displaystyle L_{ij}=x_{i}p_{j}-x_{j}p_{i}\,.}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32e4099346e97f23ae25a0c5424c77bfeaa02cce" style="vertical-align: -1.005ex; width:18.45ex; height:2.843ex;"/></span></dd></dl><p>The angular velocity can also be defined as an antisymmetric second order tensor, with components <i>ω<sub>ij</sub></i>. The relation between the two antisymmetric tensors is given by the moment of inertia which must now be a fourth order tensor:
</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle L_{ij}=I_{ijk\ell }\omega _{k\ell }\,.}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>L</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<mo>=</mo>
<msub>
<mi>I</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>j</mi>
<mi>k</mi>
<mi>ℓ<!-- ℓ --></mi>
</mrow>
</msub>
<msub>
<mi>ω<!-- ω --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
<mi>ℓ<!-- ℓ --></mi>
</mrow>
</msub>
<mspace width="thinmathspace"></mspace>
<mo>.</mo>
</mstyle>
</mrow>
{\displaystyle L_{ij}=I_{ijk\ell }\omega _{k\ell }\,.}
</semantics>
</math></span><img alt="L_{ij}=I_{ijk\ell }\omega _{k\ell }\,." aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34c1df7926b52a248cb20e1277546a850aec2045" style="vertical-align: -1.005ex; width:14.455ex; height:2.843ex;"/></span></dd></dl><p>Again, this equation in <b>L</b> and <b>ω</b> as tensors is true in any number of dimensions. This equation also appears in the geometric algebra formalism, in which <b>L</b> and <b>ω</b> are bivectors, and the moment of inertia is a mapping between them.
</p><p>In relativistic mechanics, the relativistic angular momentum of a particle is expressed as an antisymmetric tensor of second order:
</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle M_{\alpha \beta }=X_{\alpha }P_{\beta }-X_{\beta }P_{\alpha }}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>M</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>α<!-- α --></mi>
<mi>β<!-- β --></mi>
</mrow>
</msub>
<mo>=</mo>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>α<!-- α --></mi>
</mrow>
</msub>
<msub>
<mi>P</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>β<!-- β --></mi>
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<mo>−<!-- − --></mo>
<msub>
<mi>X</mi>
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</mrow>
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<mi>P</mi>
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<mi>α<!-- α --></mi>
</mrow>
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</mstyle>
</mrow>
{\displaystyle M_{\alpha \beta }=X_{\alpha }P_{\beta }-X_{\beta }P_{\alpha }}
</semantics>
</math></span><img alt="{\displaystyle M_{\alpha \beta }=X_{\alpha }P_{\beta }-X_{\beta }P_{\alpha }}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0444666934b68983d172799a6719a3b12eb2f31" style="vertical-align: -1.005ex; width:22.169ex; height:2.843ex;"/></span></dd></dl><p>in the language of four-vectors, namely the four position <i>X</i> and the four momentum <i>P</i>, and absorbs the above <b>L</b> together with the motion of the centre of mass of the particle.
</p><p>In each of the above cases, for a system of particles, the total angular momentum is just the sum of the individual particle angular momenta, and the centre of mass is for the system.
</p>samsofihttp://www.blogger.com/profile/12794342593186572847noreply@blogger.com0tag:blogger.com,1999:blog-4126202964944401087.post-63602296038929063502021-01-20T04:06:00.009-08:002021-01-20T04:06:59.502-08:00Angular momentum in quantum mechanics<img alt="" class="thumbimage" data-file-height="527" data-file-width="666" decoding="async" height="222" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a7/Classical_angular_momentum.svg/280px-Classical_angular_momentum.svg.png" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a7/Classical_angular_momentum.svg/420px-Classical_angular_momentum.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a7/Classical_angular_momentum.svg/560px-Classical_angular_momentum.svg.png 2x" width="280"/><br/><br/><br/><p>Angular momentum in quantum mechanics differs in many profound respects from angular momentum in classical mechanics. In relativistic quantum mechanics, it differs even more, in which the above relativistic definition becomes a tensorial operator.
</p><h3><span id="Spin.2C_orbital.2C_and_total_angular_momentum"></span><span class="mw-headline" id="Spin,_orbital,_and_total_angular_momentum">Spin, orbital, and total angular momentum</span><span class="mw-editsection"><span class="mw-editsection-bracket"></span>edit<span class="mw-editsection-bracket"></span></span></h3><p>The classical definition of angular momentum as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle \mathbf {L} =\mathbf {r} \times \mathbf {p} }" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">L</mi>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">r</mi>
</mrow>
<mo>×<!-- × --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">p</mi>
</mrow>
</mstyle>
</mrow>
{\displaystyle \mathbf {L} =\mathbf {r} \times \mathbf {p} }
</semantics>
</math></span><img alt="{\displaystyle \mathbf {L} =\mathbf {r} \times \mathbf {p} }" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22179a9b81408e19de312b5fbfec30ff62cefa4a" style="vertical-align: -0.671ex; width:10.135ex; height:2.509ex;"/></span> can be carried over to quantum mechanics, by reinterpreting <b>r</b> as the quantum position operator and <b>p</b> as the quantum momentum operator. <b>L</b> is then an operator, specifically called the <i>orbital angular momentum operator</i>. The components of the angular momentum operator satisfy the commutation relations of the Lie algebra so(3). Indeed, these operators are precisely the infinitesimal action of the rotation group on the quantum Hilbert space. (See also the discussion below of the angular momentum operators as the generators of rotations.)
</p><p>However, in quantum physics, there is another type of angular momentum, called <i>spin angular momentum</i>, represented by the spin operator <b>S</b>. Almost all elementary particles have nonvanishing spin. Spin is often depicted as a particle literally spinning around an axis, but this is a misleading and inaccurate picture: spin is an intrinsic property of a particle, unrelated to any sort of motion in space and fundamentally different from orbital angular momentum. All elementary particles have a characteristic spin (possibly zero), for example electrons have "spin 1/2" (this actually means "spin ħ/2"), photons have "spin 1" (this actually means "spin ħ"), and pi-mesons have spin 0.
</p><p>Finally, there is total angular momentum <b>J</b>, which combines both the spin and orbital angular momentum of all particles and fields. (For one particle, <span class="nowrap"><b>J</b> = <b>L</b> + <b>S</b></span>.) Conservation of angular momentum applies to <b>J</b>, but not to <b>L</b> or <b>S</b>; for example, the spin–orbit interaction allows angular momentum to transfer back and forth between <b>L</b> and <b>S</b>, with the total remaining constant. Electrons and photons need not have integer-based values for total angular momentum, but can also have fractional values.
</p><p>In molecules the total angular momentum <b>F</b> is the sum of the rovibronic (orbital) angular momentum <b>N</b>, the electron spin angular momentum <b>S</b>, and the nuclear spin angular momentum <b>I</b>. For electronic singlet states the rovibronic angular momentum is denoted <b>J</b> rather than <b>N</b>. As explained by Van Vleck, the components of the molecular rovibronic angular momentum referred to molecule-fixed axes have different commutation relations from those for the components about space-fixed axes.
</p><h3><span class="mw-headline" id="Quantization">Quantization</span><span class="mw-editsection"><span class="mw-editsection-bracket"></span>edit<span class="mw-editsection-bracket"></span></span></h3><p>In quantum mechanics, angular momentum is quantized – that is, it cannot vary continuously, but only in "quantum leaps" between certain allowed values. For any system, the following restrictions on measurement results apply, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle \hbar }" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi class="MJX-variant">ℏ<!-- ℏ --></mi>
</mstyle>
</mrow>
{\displaystyle \hbar }
</semantics>
</math></span><img alt="\hbar " aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de68de3a92517953436c93b5a76461d49160cc41" style="vertical-align: -0.338ex; width:1.306ex; height:2.176ex;"/></span> is the reduced Planck constant and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle {\hat {n}}}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>n</mi>
<mo stretchy="false">^<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
</mstyle>
</mrow>
{\displaystyle {\hat {n}}}
</semantics>
</math></span><img alt="{\hat {n}}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d125dccc556f5c8b0bf98a4f3847590b3f353bd4" style="vertical-align: -0.338ex; width:1.395ex; height:2.176ex;"/></span> is any Euclidean vector such as x, y, or z:
</p><table class="wikitable">
<tbody><tr>
<td><b>If you measure...</b>
</td>
<td><b>The result can be...</b>
</td></tr>
<tr>
<td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle L_{\hat {n}}}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>L</mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>n</mi>
<mo stretchy="false">^<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
</msub>
</mstyle>
</mrow>
{\displaystyle L_{\hat {n}}}
</semantics>
</math></span><img alt="{\displaystyle L_{\hat {n}}}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b78f76128624fb10cd23012eb58a31e26541a36" style="vertical-align: -1.005ex; width:2.801ex; height:2.843ex;"/></span>
</td>
<td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle \ldots ,-2\hbar ,-\hbar ,0,\hbar ,2\hbar ,\ldots }" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo>…<!-- … --></mo>
<mo>,</mo>
<mo>−<!-- − --></mo>
<mn>2</mn>
<mi class="MJX-variant">ℏ<!-- ℏ --></mi>
<mo>,</mo>
<mo>−<!-- − --></mo>
<mi class="MJX-variant">ℏ<!-- ℏ --></mi>
<mo>,</mo>
<mn>0</mn>
<mo>,</mo>
<mi class="MJX-variant">ℏ<!-- ℏ --></mi>
<mo>,</mo>
<mn>2</mn>
<mi class="MJX-variant">ℏ<!-- ℏ --></mi>
<mo>,</mo>
<mo>…<!-- … --></mo>
</mstyle>
</mrow>
{\displaystyle \ldots ,-2\hbar ,-\hbar ,0,\hbar ,2\hbar ,\ldots }
</semantics>
</math></span><img alt="\ldots ,-2\hbar ,-\hbar ,0,\hbar ,2\hbar ,\ldots " aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b19a282a732db5c4842105f212d6b5cfa0ba8b44" style="vertical-align: -0.671ex; width:24.367ex; height:2.509ex;"/></span>
</td></tr>
<tr>
<td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle S_{\hat {n}}}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>S</mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>n</mi>
<mo stretchy="false">^<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
</msub>
</mstyle>
</mrow>
{\displaystyle S_{\hat {n}}}
</semantics>
</math></span><img alt="{\displaystyle S_{\hat {n}}}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79b82b532961ce7615933feec2b306ecad9ba832" style="vertical-align: -1.005ex; width:2.643ex; height:2.843ex;"/></span> or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle J_{\hat {n}}}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>J</mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>n</mi>
<mo stretchy="false">^<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
</msub>
</mstyle>
</mrow>
{\displaystyle J_{\hat {n}}}
</semantics>
</math></span><img alt="{\displaystyle J_{\hat {n}}}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17326aeae4fa4c823e540d305e80cc0f7f708625" style="vertical-align: -1.005ex; width:2.509ex; height:2.843ex;"/></span>
</td>
<td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle \ldots ,-{\frac {3}{2}}\hbar ,-\hbar ,-{\frac {1}{2}}\hbar ,0,{\frac {1}{2}}\hbar ,\hbar ,{\frac {3}{2}}\hbar ,\ldots }" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo>…<!-- … --></mo>
<mo>,</mo>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>3</mn>
<mn>2</mn>
</mfrac>
</mrow>
<mi class="MJX-variant">ℏ<!-- ℏ --></mi>
<mo>,</mo>
<mo>−<!-- − --></mo>
<mi class="MJX-variant">ℏ<!-- ℏ --></mi>
<mo>,</mo>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
<mi class="MJX-variant">ℏ<!-- ℏ --></mi>
<mo>,</mo>
<mn>0</mn>
<mo>,</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
<mi class="MJX-variant">ℏ<!-- ℏ --></mi>
<mo>,</mo>
<mi class="MJX-variant">ℏ<!-- ℏ --></mi>
<mo>,</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>3</mn>
<mn>2</mn>
</mfrac>
</mrow>
<mi class="MJX-variant">ℏ<!-- ℏ --></mi>
<mo>,</mo>
<mo>…<!-- … --></mo>
</mstyle>
</mrow>
{\displaystyle \ldots ,-{\frac {3}{2}}\hbar ,-\hbar ,-{\frac {1}{2}}\hbar ,0,{\frac {1}{2}}\hbar ,\hbar ,{\frac {3}{2}}\hbar ,\ldots }
</semantics>
</math></span><img alt="\ldots ,-{\frac {3}{2}}\hbar ,-\hbar ,-{\frac {1}{2}}\hbar ,0,{\frac {1}{2}}\hbar ,\hbar ,{\frac {3}{2}}\hbar ,\ldots " aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82abc8246e797f65c1206e7133672cc2a800f8bd" style="vertical-align: -1.838ex; width:36.525ex; height:5.176ex;"/></span>
</td></tr>
<tr>
<td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle {\begin{aligned}&L^{2}\\={}&L_{x}^{2}+L_{y}^{2}+L_{z}^{2}\end{aligned}}}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mtable columnalign="right left right left right left right left right left right left" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true" rowspacing="3pt">
<mtr>
<mtd></mtd>
<mtd>
<msup>
<mi>L</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
</mrow>
</mtd>
<mtd>
<msubsup>
<mi>L</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>x</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msubsup>
<mo>+</mo>
<msubsup>
<mi>L</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>y</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msubsup>
<mo>+</mo>
<msubsup>
<mi>L</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>z</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msubsup>
</mtd>
</mtr>
</mtable>
</mrow>
</mstyle>
</mrow>
{\displaystyle {\begin{aligned}&L^{2}\\={}&L_{x}^{2}+L_{y}^{2}+L_{z}^{2}\end{aligned}}}
</semantics>
</math></span><img alt="{\displaystyle {\begin{aligned}&L^{2}\\={}&L_{x}^{2}+L_{y}^{2}+L_{z}^{2}\end{aligned}}}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1142965311188c2ec11536bcd0a200235f145a96" style="vertical-align: -2.505ex; width:16.915ex; height:6.176ex;"/></span>
</td>
<td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle \left\hbar ^{2}n(n+1)\right}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow>
<mo></mo>
<mrow>
<msup>
<mi class="MJX-variant">ℏ<!-- ℏ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mi>n</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
<mo></mo>
</mrow>
</mstyle>
</mrow>
{\displaystyle \left\hbar ^{2}n(n+1)\right}
</semantics>
</math></span><img alt="{\displaystyle \left\hbar ^{2}n(n+1)\right}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd720bdca13e8588818fb477347fb90c9789bb4a" style="vertical-align: -1.005ex; width:12.917ex; height:3.343ex;"/></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle n=0,1,2,\ldots }" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
<mo>=</mo>
<mn>0</mn>
<mo>,</mo>
<mn>1</mn>
<mo>,</mo>
<mn>2</mn>
<mo>,</mo>
<mo>…<!-- … --></mo>
</mstyle>
</mrow>
{\displaystyle n=0,1,2,\ldots }
</semantics>
</math></span><img alt="{\displaystyle n=0,1,2,\ldots }" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a19cb2cfd4f9ebdbc8e5cbb9b92ecb9ace85cab" style="vertical-align: -0.671ex; width:13.806ex; height:2.509ex;"/></span>
</td></tr>
<tr>
<td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle S^{2}}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>S</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
{\displaystyle S^{2}}
</semantics>
</math></span><img alt="S^{2}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b6401d5d0155afb1406770d1eb80badce4e08ce" style="vertical-align: -0.338ex; width:2.576ex; height:2.676ex;"/></span> or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle J^{2}}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>J</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
{\displaystyle J^{2}}
</semantics>
</math></span><img alt="J^{2}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6fa18f680f853f4b3462401d4012c9b430b88b4f" style="vertical-align: -0.338ex; width:2.58ex; height:2.676ex;"/></span>
</td>
<td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle \left\hbar ^{2}n(n+1)\right}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow>
<mo></mo>
<mrow>
<msup>
<mi class="MJX-variant">ℏ<!-- ℏ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mi>n</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
<mo></mo>
</mrow>
</mstyle>
</mrow>
{\displaystyle \left\hbar ^{2}n(n+1)\right}
</semantics>
</math></span><img alt="{\displaystyle \left\hbar ^{2}n(n+1)\right}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd720bdca13e8588818fb477347fb90c9789bb4a" style="vertical-align: -1.005ex; width:12.917ex; height:3.343ex;"/></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle n=0,{\frac {1}{2}},1,{\frac {3}{2}},\ldots }" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
<mo>=</mo>
<mn>0</mn>
<mo>,</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
<mo>,</mo>
<mn>1</mn>
<mo>,</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>3</mn>
<mn>2</mn>
</mfrac>
</mrow>
<mo>,</mo>
<mo>…<!-- … --></mo>
</mstyle>
</mrow>
{\displaystyle n=0,{\frac {1}{2}},1,{\frac {3}{2}},\ldots }
</semantics>
</math></span><img alt="{\displaystyle n=0,{\frac {1}{2}},1,{\frac {3}{2}},\ldots }" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a7134cb22605909535e6ca123a2d7aaa2afd8e3" style="vertical-align: -1.838ex; width:17.674ex; height:5.176ex;"/></span>
</td></tr></tbody></table><p>(There are additional restrictions as well, see angular momentum operator for details.)
</p><p>The reduced Planck constant <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle \hbar }" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi class="MJX-variant">ℏ<!-- ℏ --></mi>
</mstyle>
</mrow>
{\displaystyle \hbar }
</semantics>
</math></span><img alt="\hbar " aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de68de3a92517953436c93b5a76461d49160cc41" style="vertical-align: -0.338ex; width:1.306ex; height:2.176ex;"/></span> is tiny by everyday standards, about 10−34 J s, and therefore this quantization does not noticeably affect the angular momentum of macroscopic objects. However, it is very important in the microscopic world. For example, the structure of electron shells and subshells in chemistry is significantly affected by the quantization of angular momentum.
</p><p>Quantization of angular momentum was first postulated by Niels Bohr in his Bohr model of the atom and was later predicted by Erwin Schrödinger in his Schrödinger equation.
</p><h3><span class="mw-headline" id="Uncertainty">Uncertainty</span><span class="mw-editsection"><span class="mw-editsection-bracket"></span>edit<span class="mw-editsection-bracket"></span></span></h3><p>In the definition <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle \mathbf {L} =\mathbf {r} \times \mathbf {p} }" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">L</mi>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">r</mi>
</mrow>
<mo>×<!-- × --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">p</mi>
</mrow>
</mstyle>
</mrow>
{\displaystyle \mathbf {L} =\mathbf {r} \times \mathbf {p} }
</semantics>
</math></span><img alt="\mathbf {L} =\mathbf {r} \times \mathbf {p} " aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22179a9b81408e19de312b5fbfec30ff62cefa4a" style="vertical-align: -0.671ex; width:10.135ex; height:2.509ex;"/></span>, six operators are involved: The position operators <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle r_{x}}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>r</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>x</mi>
</mrow>
</msub>
</mstyle>
</mrow>
{\displaystyle r_{x}}
</semantics>
</math></span><img alt="r_{x}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3705946c668b0bd4b21e7c19b95bb6ed05ea4ed" style="vertical-align: -0.671ex; width:2.221ex; height:2.009ex;"/></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle r_{y}}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>r</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>y</mi>
</mrow>
</msub>
</mstyle>
</mrow>
{\displaystyle r_{y}}
</semantics>
</math></span><img alt="r_{y}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49e52760272851bdfd60c01877ee11bfc1857a60" style="vertical-align: -1.005ex; width:2.098ex; height:2.343ex;"/></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle r_{z}}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>r</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>z</mi>
</mrow>
</msub>
</mstyle>
</mrow>
{\displaystyle r_{z}}
</semantics>
</math></span><img alt="r_{z}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07006192648caeccd5ad603f234b1f11e2e40b9" style="vertical-align: -0.671ex; width:2.05ex; height:2.009ex;"/></span>, and the momentum operators <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle p_{x}}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>x</mi>
</mrow>
</msub>
</mstyle>
</mrow>
{\displaystyle p_{x}}
</semantics>
</math></span><img alt="p_{x}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a5055ed65713825b48aa6ee05118c072e6f026a" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.431ex; height:2.009ex;"/></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle p_{y}}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>y</mi>
</mrow>
</msub>
</mstyle>
</mrow>
{\displaystyle p_{y}}
</semantics>
</math></span><img alt="p_{y}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1108b3eff778ca6f026e3e28c26cb093174cc2d0" style="vertical-align: -1.005ex; margin-left: -0.089ex; width:2.308ex; height:2.343ex;"/></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle p_{z}}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>z</mi>
</mrow>
</msub>
</mstyle>
</mrow>
{\displaystyle p_{z}}
</semantics>
</math></span><img alt="p_{z}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1dbd3c1a6173a7974e0095301da94447c5f67657" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.261ex; height:2.009ex;"/></span>. However, the Heisenberg uncertainty principle tells us that it is not possible for all six of these quantities to be known simultaneously with arbitrary precision. Therefore, there are limits to what can be known or measured about a particle's angular momentum. It turns out that the best that one can do is to simultaneously measure both the angular momentum vector's magnitude and its component along one axis.
</p><p>The uncertainty is closely related to the fact that different components of an angular momentum operator do not commute, for example <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle L_{x}L_{y}\neq L_{y}L_{x}}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>L</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>x</mi>
</mrow>
</msub>
<msub>
<mi>L</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>y</mi>
</mrow>
</msub>
<mo>≠<!-- ≠ --></mo>
<msub>
<mi>L</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>y</mi>
</mrow>
</msub>
<msub>
<mi>L</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>x</mi>
</mrow>
</msub>
</mstyle>
</mrow>
{\displaystyle L_{x}L_{y}\neq L_{y}L_{x}}
</semantics>
</math></span><img alt="L_{x}L_{y}\neq L_{y}L_{x}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14b5e35b0c4b63f2a2bd24c39f08e6e1c86f9528" style="vertical-align: -1.005ex; width:13.873ex; height:2.843ex;"/></span>. (For the precise commutation relations, see angular momentum operator.)
</p><h3><span class="mw-headline" id="Total_angular_momentum_as_generator_of_rotations">Total angular momentum as generator of rotations</span><span class="mw-editsection"><span class="mw-editsection-bracket"></span>edit<span class="mw-editsection-bracket"></span></span></h3><p>As mentioned above, orbital angular momentum <b>L</b> is defined as in classical mechanics: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle \mathbf {L} =\mathbf {r} \times \mathbf {p} }" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">L</mi>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">r</mi>
</mrow>
<mo>×<!-- × --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">p</mi>
</mrow>
</mstyle>
</mrow>
{\displaystyle \mathbf {L} =\mathbf {r} \times \mathbf {p} }
</semantics>
</math></span><img alt="\mathbf {L} =\mathbf {r} \times \mathbf {p} " aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22179a9b81408e19de312b5fbfec30ff62cefa4a" style="vertical-align: -0.671ex; width:10.135ex; height:2.509ex;"/></span>, but <i>total</i> angular momentum <b>J</b> is defined in a different, more basic way: <b>J</b> is defined as the "generator of rotations". More specifically, <b>J</b> is defined so that the operator
</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle R({\hat {n}},\phi )\equiv \exp \left(-{\frac {i}{\hbar }}\phi \,\mathbf {J} \cdot {\hat {\mathbf {n} }}\right)}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>R</mi>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>n</mi>
<mo stretchy="false">^<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
<mo>,</mo>
<mi>ϕ<!-- ϕ --></mi>
<mo stretchy="false">)</mo>
<mo>≡<!-- ≡ --></mo>
<mi>exp</mi>
<mo><!-- --></mo>
<mrow>
<mo>(</mo>
<mrow>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>i</mi>
<mi class="MJX-variant">ℏ<!-- ℏ --></mi>
</mfrac>
</mrow>
<mi>ϕ<!-- ϕ --></mi>
<mspace width="thinmathspace"></mspace>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">J</mi>
</mrow>
<mo>⋅<!-- ⋅ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">n</mi>
</mrow>
<mo stretchy="false">^<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</mstyle>
</mrow>
{\displaystyle R({\hat {n}},\phi )\equiv \exp \left(-{\frac {i}{\hbar }}\phi \,\mathbf {J} \cdot {\hat {\mathbf {n} }}\right)}
</semantics>
</math></span><img alt="R({\hat {n}},\phi )\equiv \exp \left(-{\frac {i}{\hbar }}\phi \,\mathbf {J} \cdot {\hat {\mathbf {n} }}\right)" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05d8a19780700ce3d93b241a525c4441f6479e34" style="vertical-align: -2.505ex; width:27.728ex; height:6.176ex;"/></span></dd></dl><p>is the rotation operator that takes any system and rotates it by angle <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle \phi }" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ϕ<!-- ϕ --></mi>
</mstyle>
</mrow>
{\displaystyle \phi }
</semantics>
</math></span><img alt="\phi " aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;"/></span> about the axis <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle {\hat {\mathbf {n} }}}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">n</mi>
</mrow>
<mo stretchy="false">^<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
</mstyle>
</mrow>
{\displaystyle {\hat {\mathbf {n} }}}
</semantics>
</math></span><img alt="{\hat {\mathbf {n} }}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aae87b164ba005e99b51066c46d1eacc7f56564a" style="vertical-align: -0.338ex; width:1.485ex; height:2.343ex;"/></span>. (The "exp" in the formula refers to operator exponential) To put this the other way around, whatever our quantum Hilbert space is, we expect that the rotation group SO(3) will act on it. There is then an associated action of the Lie algebra so(3) of SO(3); the operators describing the action of so(3) on our Hilbert space are the (total) angular momentum operators.
</p><p>The relationship between the angular momentum operator and the rotation operators is the same as the relationship between Lie algebras and Lie groups in mathematics. The close relationship between angular momentum and rotations is reflected in Noether's theorem that proves that angular momentum is conserved whenever the laws of physics are rotationally invariant.
</p>samsofihttp://www.blogger.com/profile/12794342593186572847noreply@blogger.com0tag:blogger.com,1999:blog-4126202964944401087.post-43432188031226604252021-01-20T04:06:00.007-08:002021-01-20T04:06:54.672-08:00Angular momentum in electrodynamics<img alt="" class="thumbimage" data-file-height="256" data-file-width="256" decoding="async" height="256" src="//upload.wikimedia.org/wikipedia/commons/7/73/Newton_area_law_derivation.gif" width="256"/><br/><br/><br/><p>When describing the motion of a charged particle in an electromagnetic field, the canonical momentum <b>P</b> (derived from the Lagrangian for this system) is not gauge invariant. As a consequence, the canonical angular momentum <b>L</b> = <b>r</b> × <b>P</b> is not gauge invariant either. Instead, the momentum that is physical, the so-called <i>kinetic momentum</i> (used throughout this article), is (in SI units)
</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle \mathbf {p} =m\mathbf {v} =\mathbf {P} -e\mathbf {A} }" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">p</mi>
</mrow>
<mo>=</mo>
<mi>m</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">v</mi>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">P</mi>
</mrow>
<mo>−<!-- − --></mo>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">A</mi>
</mrow>
</mstyle>
</mrow>
{\displaystyle \mathbf {p} =m\mathbf {v} =\mathbf {P} -e\mathbf {A} }
</semantics>
</math></span><img alt="\mathbf {p} =m\mathbf {v} =\mathbf {P} -e\mathbf {A} " aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b13cdc37559fc7edc45f19ca88d0763bb4e5000" style="vertical-align: -0.671ex; width:18.904ex; height:2.509ex;"/></span></dd></dl><p>where <i>e</i> is the electric charge of the particle and <b>A</b> the magnetic vector potential of the electromagnetic field. The gauge-invariant angular momentum, that is <i>kinetic angular momentum</i>, is given by
</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle \mathbf {K} =\mathbf {r} \times (\mathbf {P} -e\mathbf {A} )}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">K</mi>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">r</mi>
</mrow>
<mo>×<!-- × --></mo>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">P</mi>
</mrow>
<mo>−<!-- − --></mo>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">A</mi>
</mrow>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
{\displaystyle \mathbf {K} =\mathbf {r} \times (\mathbf {P} -e\mathbf {A} )}
</semantics>
</math></span><img alt="\mathbf {K} =\mathbf {r} \times (\mathbf {P} -e\mathbf {A} )" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ea5e7e0c75ea3090123ba971034a6cd72db6b79" style="vertical-align: -0.838ex; width:18.714ex; height:2.843ex;"/></span></dd></dl><p>The interplay with quantum mechanics is discussed further in the article on canonical commutation relations.
</p>samsofihttp://www.blogger.com/profile/12794342593186572847noreply@blogger.com0tag:blogger.com,1999:blog-4126202964944401087.post-11632960752501526072021-01-20T04:06:00.005-08:002021-01-20T04:06:50.983-08:00Angular momentum in optics<img alt="" class="thumbimage" data-file-height="256" data-file-width="256" decoding="async" height="256" src="//upload.wikimedia.org/wikipedia/commons/7/73/Newton_area_law_derivation.gif" width="256"/><br/><br/><br/><p>In <i>classical Maxwell electrodynamics</i> the Poynting vector
is a linear momentum density of electromagnetic field.
</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle \mathbf {S} (\mathbf {r} ,t)=\epsilon _{0}c^{2}\mathbf {E} (\mathbf {r} ,t)\times \mathbf {B} (\mathbf {r} ,t).}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">S</mi>
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<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">r</mi>
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<mo>,</mo>
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<mo stretchy="false">)</mo>
<mo>=</mo>
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<mi>ϵ<!-- ϵ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
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<msup>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
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</msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">E</mi>
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<mo stretchy="false">(</mo>
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<mo stretchy="false">)</mo>
<mo>×<!-- × --></mo>
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<mi mathvariant="bold">B</mi>
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<mo stretchy="false">(</mo>
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<mo>,</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mo>.</mo>
</mstyle>
</mrow>
{\displaystyle \mathbf {S} (\mathbf {r} ,t)=\epsilon _{0}c^{2}\mathbf {E} (\mathbf {r} ,t)\times \mathbf {B} (\mathbf {r} ,t).}
</semantics>
</math></span><img alt="{\displaystyle \mathbf {S} (\mathbf {r} ,t)=\epsilon _{0}c^{2}\mathbf {E} (\mathbf {r} ,t)\times \mathbf {B} (\mathbf {r} ,t).}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1def5c88bdb99e90ec83e577815c9a68fdf6b4e" style="vertical-align: -0.838ex; width:30.143ex; height:3.176ex;"/></span></dd></dl><p>The angular momentum density vector <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle \mathbf {L} (\mathbf {r} ,t)}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">L</mi>
</mrow>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">r</mi>
</mrow>
<mo>,</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
{\displaystyle \mathbf {L} (\mathbf {r} ,t)}
</semantics>
</math></span><img alt="{\displaystyle \mathbf {L} (\mathbf {r} ,t)}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dac2bad85a420ba827c3449d04bfd020224dacad" style="vertical-align: -0.838ex; width:6.393ex; height:2.843ex;"/></span> is given by a vector product
as in classical mechanics:
</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle \mathbf {L} (\mathbf {r} ,t)=\mathbf {r} \times \mathbf {S} (\mathbf {r} ,t).}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">L</mi>
</mrow>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">r</mi>
</mrow>
<mo>,</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">r</mi>
</mrow>
<mo>×<!-- × --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">S</mi>
</mrow>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">r</mi>
</mrow>
<mo>,</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mo>.</mo>
</mstyle>
</mrow>
{\displaystyle \mathbf {L} (\mathbf {r} ,t)=\mathbf {r} \times \mathbf {S} (\mathbf {r} ,t).}
</semantics>
</math></span><img alt="{\displaystyle \mathbf {L} (\mathbf {r} ,t)=\mathbf {r} \times \mathbf {S} (\mathbf {r} ,t).}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9f27863b4969ddd4058a037822bd0033ad8de30" style="vertical-align: -0.838ex; width:20.351ex; height:2.843ex;"/></span></dd></dl><p>The above identities are valid <i>locally</i>, i.e. in each space point <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle \mathbf {r} }" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">r</mi>
</mrow>
</mstyle>
</mrow>
{\displaystyle \mathbf {r} }
</semantics>
</math></span><img alt="\mathbf {r} " aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eca0f46511c4c986c48b254073732c0bd98ae0c1" style="vertical-align: -0.338ex; width:1.102ex; height:1.676ex;"/></span> in a given moment <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math alttext="{\displaystyle t}" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>t</mi>
</mstyle>
</mrow>
{\displaystyle t}
</semantics>
</math></span><img alt="t" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;"/></span>.
</p>samsofihttp://www.blogger.com/profile/12794342593186572847noreply@blogger.com0tag:blogger.com,1999:blog-4126202964944401087.post-24131726248320526592021-01-20T04:06:00.003-08:002021-01-20T04:06:47.101-08:00History<img alt="" class="thumbimage" data-file-height="256" data-file-width="256" decoding="async" height="256" src="//upload.wikimedia.org/wikipedia/commons/7/73/Newton_area_law_derivation.gif" width="256"/><br/><br/><br/><p>Newton, in the <i>Principia</i>, hinted at angular momentum in his examples of the First Law of Motion,
</p><dl><dd><i>A top, whose parts by their cohesion are perpetually drawn aside from rectilinear motions, does not cease its rotation, otherwise than as it is retarded by the air. The greater bodies of the planets and comets, meeting with less resistance in more free spaces, preserve their motions both progressive and circular for a much longer time.</i></dd></dl><p>He did not further investigate angular momentum directly in the <i>Principia</i>,
</p><dl><dd><i>From such kind of reflexions also sometimes arise the circular motions of bodies about their own centres. But these are cases which I do not consider in what follows; and it would be too tedious to demonstrate every particular that relates to this subject.</i></dd></dl><p>However, his geometric proof of the law of areas is an outstanding example of Newton's genius, and indirectly proves angular momentum conservation in the case of a central force.
</p><h3><span class="mw-headline" id="The_Law_of_Areas">The Law of Areas</span><span class="mw-editsection"><span class="mw-editsection-bracket"></span>edit<span class="mw-editsection-bracket"></span></span></h3><h4><span id="Newton.27s_derivation"></span><span class="mw-headline" id="Newton's_derivation">Newton's derivation</span><span class="mw-editsection"><span class="mw-editsection-bracket"></span>edit<span class="mw-editsection-bracket"></span></span></h4><p>As a planet orbits the Sun, the line between the Sun and the planet sweeps out equal areas in equal intervals of time. This had been known since Kepler expounded his second law of planetary motion. Newton derived a unique geometric proof, and went on to show that the attractive force of the Sun's gravity was the cause of all of Kepler's laws.
</p><p>During the first interval of time, an object is in motion from point <b>A</b> to point <b>B</b>. Undisturbed, it would continue to point <b>c</b> during the second interval. When the object arrives at <b>B</b>, it receives an impulse directed toward point <b>S</b>. The impulse gives it a small added velocity toward <b>S</b>, such that if this were its only velocity, it would move from <b>B</b> to <b>V</b> during the second interval. By the rules of velocity composition, these two velocities add, and point <b>C</b> is found by construction of parallelogram <b>BcCV</b>. Thus the object's path is deflected by the impulse so that it arrives at point <b>C</b> at the end of the second interval. Because the triangles <b>SBc</b> and <b>SBC</b> have the same base <b>SB</b> and the same height <b>Bc</b> or <b>VC</b>, they have the same area. By symmetry, triangle <b>SBc</b> also has the same area as triangle <b>SAB</b>, therefore the object has swept out equal areas <b>SAB</b> and <b>SBC</b> in equal times.
</p><p>At point <b>C</b>, the object receives another impulse toward <b>S</b>, again deflecting its path during the third interval from <b>d</b> to <b>D</b>. Thus it continues to <b>E</b> and beyond, the triangles <b>SAB</b>, <b>SBc</b>, <b>SBC</b>, <b>SCd</b>, <b>SCD</b>, <b>SDe</b>, <b>SDE</b> all having the same area. Allowing the time intervals to become ever smaller, the path <b>ABCDE</b> approaches indefinitely close to a continuous curve.
</p><p>Note that because this derivation is geometric, and no specific force is applied, it proves a more general law than Kepler's second law of planetary motion. It shows that the Law of Areas applies to any central force, attractive or repulsive, continuous or non-continuous, or zero.
</p><h4><span class="mw-headline" id="Conservation_of_angular_momentum_in_the_Law_of_Areas">Conservation of angular momentum in the Law of Areas</span><span class="mw-editsection"><span class="mw-editsection-bracket"></span>edit<span class="mw-editsection-bracket"></span></span></h4><p>The proportionality of angular momentum to the area swept out by a moving object can be understood by realizing that the bases of the triangles, that is, the lines from <b>S</b> to the object, are equivalent to the radius <span class="texhtml"><var>r</var></span>, and that the heights of the triangles are proportional to the perpendicular component of velocity <span class="texhtml"><var>v</var><sub>⊥</sub></span>. Hence, if the area swept per unit time is constant, then by the triangular area formula <span class="texhtml"><style data-mw-deduplicate="TemplateStyles:r993651011">.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px;white-space:nowrap}</style><span class="sfrac nowrap tion" role="math" style="display:inline-block; vertical-align:-0.5em; font-size:85%; text-align:center;"><span class="num" style="display:block; line-height:1em; margin:0 0.1em;">1</span><span class="slash sr-only">/</span><span class="den" style="display:block; line-height:1em; margin:0 0.1em; border-top:1px solid;">2</span></span>(base)(height)</span>, the product <span class="texhtml">(base)(height)</span> and therefore the product <span class="texhtml"><var>rv</var><sub>⊥</sub></span> are constant: if <span class="texhtml"><var>r</var></span> and the base length are decreased, <span class="texhtml"><var>v</var><sub>⊥</sub></span> and height must increase proportionally. Mass is constant, therefore angular momentum <span class="texhtml"><var>rmv</var><sub>⊥</sub></span> is conserved by this exchange of distance and velocity.
</p><p>In the case of triangle <b>SBC</b>, area is equal to <link href="mw-data:TemplateStyles:r993651011" rel="mw-deduplicated-inline-style"/><span class="sfrac nowrap tion" role="math" style="display:inline-block; vertical-align:-0.5em; font-size:85%; text-align:center;"><span class="num" style="display:block; line-height:1em; margin:0 0.1em;">1</span><span class="slash sr-only">/</span><span class="den" style="display:block; line-height:1em; margin:0 0.1em; border-top:1px solid;">2</span></span>(<b>SB</b>)(<b>VC</b>). Wherever <b>C</b> is eventually located due to the impulse applied at <b>B</b>, the product (<b>SB</b>)(<b>VC</b>), and therefore <span class="texhtml"><var>rmv</var><sub>⊥</sub></span> remain constant. Similarly so for each of the triangles.
</p><h3><span class="mw-headline" id="After_Newton">After Newton</span><span class="mw-editsection"><span class="mw-editsection-bracket"></span>edit<span class="mw-editsection-bracket"></span></span></h3><p>Leonhard Euler, Daniel Bernoulli, and Patrick d'Arcy all understood angular momentum in terms of conservation of areal velocity, a result of their analysis of Kepler's second law of planetary motion. It is unlikely that they realized the implications for ordinary rotating matter.
</p><p>In 1736 Euler, like Newton, touched on some of the equations of angular momentum in his <i>Mechanica</i> without further developing them.
</p><p>Bernoulli wrote in a 1744 letter of a "moment of rotational motion", possibly the first conception of angular momentum as we now understand it.
</p><p>In 1799, Pierre-Simon Laplace first realized that a fixed plane was associated with rotation—his <i>invariable plane</i>.
</p><p>Louis Poinsot in 1803 began representing rotations as a line segment perpendicular to the rotation, and elaborated on the "conservation of moments".
</p><p>In 1852 Léon Foucault used a gyroscope in an experiment to display the Earth's rotation.
</p><p>William J. M. Rankine's 1858 <i>Manual of Applied Mechanics</i> defined angular momentum in the modern sense for the first time:
</p><dl><dd><i>...a line whose length is proportional to the magnitude of the angular momentum, and whose direction is perpendicular to the plane of motion of the body and of the fixed point, and such, that when the motion of the body is viewed from the extremity of the line, the radius-vector of the body seems to have right-handed rotation.</i></dd></dl><p>In an 1872 edition of the same book, Rankine stated that "The term <i>angular momentum</i> was introduced by Mr. Hayward," probably referring to R.B. Hayward's article <i>On a Direct Method of estimating Velocities, Accelerations, and all similar Quantities with respect to Axes moveable in any manner in Space with Applications,</i> which was introduced in 1856, and published in 1864. Rankine was mistaken, as numerous publications feature the term starting in the late 18th to early 19th centuries. However, Hayward's article apparently was the first use of the term and the concept seen by much of the English-speaking world. Before this, angular momentum was typically referred to as "momentum of rotation" in English.
</p>samsofihttp://www.blogger.com/profile/12794342593186572847noreply@blogger.com0tag:blogger.com,1999:blog-4126202964944401087.post-89508973296525540952021-01-20T04:06:00.001-08:002021-01-20T04:06:43.218-08:00Footnotes<br/><br/><br/>samsofihttp://www.blogger.com/profile/12794342593186572847noreply@blogger.com0