Editing Momentum (section)

==Generalized==
{{See also|Analytical mechanics}}
Newton's laws can be difficult to apply to many kinds of motion because the motion is limited by ''constraints''. For example, a bead on an abacus is constrained to move along its wire and a pendulum bob is constrained to swing at a fixed distance from the pivot. Many such constraints can be incorporated by changing the normal [[Cartesian coordinates]] to a set of ''[[generalized coordinates]]'' that may be fewer in number.<ref>{{harvnb|Goldstein|1980|pp=11–13}}</ref> Refined mathematical methods have been developed for solving mechanics problems in generalized coordinates. They introduce a ''generalized momentum'', also known as the ''canonical momentum'' or ''conjugate momentum'', that extends the concepts of both linear momentum and [[angular momentum]]. To distinguish it from generalized momentum, the product of mass and velocity is also referred to as ''mechanical momentum'', ''kinetic momentum'' or ''kinematic momentum''.<ref name=Goldstein54>{{harvnb|Goldstein|1980|pp=54–56}}</ref><ref>{{harvnb|Jackson|1975|p=574}}</ref><ref name=FeynmanQM>[https://feynmanlectures.caltech.edu/III_21.html#Ch21-S3 ''The Feynman Lectures on Physics''] Vol. III Ch. 21-3: Two kinds of momentum</ref> The two main methods are described below.

===Lagrangian mechanics===
In [[Lagrangian mechanics]], a Lagrangian is defined as the difference between the kinetic energy {{mvar|T}} and the [[potential energy]] {{mvar|V}}:

<math display="block"> \mathcal{L}  = T-V\,.</math>

If the generalized coordinates are represented as a vector {{math|'''q''' {{=}} ({{var|q}}{{sub|1}}, {{var|q}}{{sub|2}}, ... , {{var|q}}{{sub|{{var|N}}}}) }} and time differentiation is represented by a dot over the variable, then the equations of motion (known as the Lagrange or [[Euler–Lagrange equation]]s) are a set of {{mvar|N}} equations:<ref>{{harvnb|Goldstein|1980|pp=20–21}}</ref>

<math display="block"> \frac{\text{d}}{\text{d}t}\left(\frac{\partial \mathcal{L} }{\partial\dot{q}_j}\right) - \frac{\partial \mathcal{L}}{\partial q_j} = 0\,.</math>

If a coordinate {{math|{{var|q}}{{sub|{{var|i}}}}}} is not a Cartesian coordinate, the associated generalized momentum component {{math|{{var|p}}{{sub|{{var|i}}}}}} does not necessarily have the dimensions of linear momentum. Even if {{math|{{var|q}}{{sub|{{var|i}}}}}} is a Cartesian coordinate, {{math|{{var|p}}{{sub|{{var|i}}}}}} will not be the same as the mechanical momentum if the potential depends on velocity.<ref name=Goldstein54/> Some sources represent the kinematic momentum by the symbol {{math|'''Π'''}}.<ref name=Lerner>{{cite book|editor-last=Lerner|editor-first=Rita G.|editor-link=Rita G. Lerner|title=Encyclopedia of Physics|date=2005|publisher=Wiley-VCH|location=Weinheim|isbn=978-3-527-40554-1|edition=3rd |editor2-last=Trigg |editor2-first=George L.}}</ref>

In this mathematical framework, a generalized momentum is associated with the generalized coordinates. Its components are defined as

<math display="block"> p_j = \frac{\partial \mathcal{L} }{\partial \dot{q}_j}\,.</math>

Each component {{math|{{var|p}}{{sub|{{var|j}}}}}} is said to be the ''conjugate momentum'' for the coordinate {{math|{{var|q}}{{sub|{{var|j}}}}}}.

Now if a given coordinate {{math|{{var|q}}{{sub|{{var|i}}}}}} does not appear in the Lagrangian (although its time derivative might appear), then {{math|{{var|p}}{{sub|{{var|j}}}}}} is constant. This is the generalization of the conservation of momentum.<ref name=Goldstein54/>

Even if the generalized coordinates are just the ordinary spatial coordinates, the conjugate momenta are not necessarily the ordinary momentum coordinates. An example is found in the section on electromagnetism.

===Hamiltonian mechanics===
In [[Hamiltonian mechanics]], the Lagrangian (a function of generalized coordinates and their derivatives) is replaced by a Hamiltonian that is a function of generalized coordinates and momentum. The Hamiltonian is defined as

<math display="block"> \mathcal{H}\left(\mathbf{q},\mathbf{p},t\right) = \mathbf{p}\cdot\dot{\mathbf{q}} - \mathcal{L}\left(\mathbf{q},\dot{\mathbf{q}},t\right)\,,</math>

where the momentum is obtained by differentiating the Lagrangian as above. The Hamiltonian equations of motion are<ref>{{harvnb|Goldstein|1980|pp=341–342}}</ref>

<math display="block"> \begin{align}
\dot{q}_i &= \frac{\partial\mathcal{H}}{\partial p_i}\\
-\dot{p}_i &= \frac{\partial\mathcal{H}}{\partial q_i}\\
-\frac{\partial \mathcal{L}}{\partial t} &= \frac{\text{d} \mathcal{H}}{\text{d}t}\,.
\end{align}</math>

As in Lagrangian mechanics, if a generalized coordinate does not appear in the Hamiltonian, its conjugate momentum component is conserved.<ref>{{harvnb|Goldstein|1980|p=348}}</ref>

===Symmetry and conservation===
Conservation of momentum is a mathematical consequence of the [[Homogeneity (physics)|homogeneity]] (shift [[symmetry]]) of space (position in space is the [[canonical conjugate]] quantity to momentum). That is, conservation of momentum is a consequence of the fact that the laws of physics do not depend on position; this is a special case of [[Noether's theorem]].<ref>{{cite book|last1=Hand|first1=Louis N. |last2=Finch |first2=Janet D. |title=Analytical mechanics|date=1998|publisher=Cambridge University Press|location=Cambridge|isbn=978-0-521-57572-0|edition=7th print |pages=Chapter 4 |no-pp=true}}</ref> For systems that do not have this symmetry, it may not be possible to define conservation of momentum. Examples where conservation of momentum does not apply include [[curved space]]times in [[general relativity]]<ref>{{cite journal|last1=Witten|first1=Edward|title=A new proof of the positive energy theorem|journal=Communications in Mathematical Physics|volume=80|issue=3|year=1981|pages=381–402|issn=0010-3616|doi=10.1007/BF01208277|bibcode=1981CMaPh..80..381W|s2cid=1035111|url=https://www.sns.ias.edu/ckfinder/userfiles/files/%5B32%5DCMP_80_1981.pdf|access-date=2020-12-17|archive-date=2016-11-25|archive-url=https://web.archive.org/web/20161125044504/https://www.sns.ias.edu/ckfinder/userfiles/files/%5B32%5DCMP_80_1981.pdf}}</ref> or [[time crystals]] in [[condensed matter physics]].<ref name="Grossman 2012">{{cite magazine|last1=Grossman|first1=Lisa|title=Death-defying time crystal could outlast the universe|url=https://www.newscientist.com/article/mg21328484-000-death-defying-time-crystal-could-outlast-the-universe/|magazine=New Scientist|archive-url=https://archive.today/20170202104619/https://www.newscientist.com/article/mg21328484-000-death-defying-time-crystal-could-outlast-the-universe/|archive-date=2017-02-02|date=18 January 2012}}</ref><ref name="Cowen 2012">{{cite magazine|last1=Cowen|first1=Ron|title='Time Crystals' Could Be a Legitimate Form of Perpetual Motion|url=https://www.scientificamerican.com/article/time-crystals-could-be-legitimate-form-perpetual-motion/|magazine=Scientific American|archive-url=https://archive.today/20170202101455/https://www.scientificamerican.com/article/time-crystals-could-be-legitimate-form-perpetual-motion/|archive-date=2017-02-02|date=27 February 2012}}</ref><ref name="Powell 2013">{{cite journal|last1=Powell|first1=Devin|title=Can matter cycle through shapes eternally?|journal=Nature|year=2013|issn=1476-4687|doi=10.1038/nature.2013.13657|s2cid=181223762|url=http://www.nature.com/news/can-matter-cycle-through-shapes-eternally-1.13657|archive-url=https://archive.today/20170203080014/http://www.nature.com/news/can-matter-cycle-through-shapes-eternally-1.13657|archive-date=2017-02-03}}</ref><ref name="Gibney 2017">{{cite journal|last1=Gibney|first1=Elizabeth|title=The quest to crystallize time|journal=Nature|volume=543|issue=7644|year=2017|pages=164–166|issn=0028-0836|doi=10.1038/543164a|pmid=28277535|url=http://www.nature.com/news/the-quest-to-crystallize-time-1.21595|archive-url=https://archive.today/20170313115721/http://www.nature.com/news/the-quest-to-crystallize-time-1.21595|archive-date=2017-03-13|bibcode=2017Natur.543..164G|s2cid=4460265}}</ref>