Kinetic momentum: Difference between revisions
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#REDIRECT [[momentum]] |
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{{mergeto|Momentum|discuss=Talk:Momentum#Merger proposal|date=July 2012}} |
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{{hatnote|[[Potential momentum]] redirects here. See the main article for [[momentum]] in general.}} |
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In [[physics]], in particular [[electromagnetism]], the '''kinetic momentum''' is the [[momentum]] |
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:<math>\mathbf{p} = m\mathbf{v}</math> |
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of a [[charged particle]] of mass ''m'' and [[velocity]] '''v''' moving in an [[electromagnetic field]]. |
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==Non-relativistic dynamics== |
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===Lagrangian formulation=== |
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Kinetic momentum is distinct from the ''[[canonical momentum]]'' encountered in [[Lagrangian mechanics]]. The Lagrangian, ''L'', of a free charged particle is |
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:<math>L=\frac{m}{2}\mathbf{\dot{r}}\cdot\mathbf{\dot{r}}+e\mathbf{A}\cdot\mathbf{\dot{r}}-e\phi,</math> |
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where ''m'' is the particle's mass, ''e'' is its charge, <math>\mathbf{\dot{r}} = \mathbf{v}</math> is its [[velocity]], and φ and '''A''' are, respectively, the [[electric potential|scalar potential]] and the [[magnetic vector potential|vector potential]] of the electromagnetic field. The canonical momentum is defined as |
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:<math>\mathbf{P} = \frac{\partial L}{\partial \mathbf{\dot{r}}} = m\mathbf{\dot{r}} +e\mathbf{A},</math> |
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and hence the kinetic momentum is related to the canonical momentum by<ref name=Lerner>{{cite book|editor-last=Lerner|editor-first=Rita G.|title=Encyclopedia of physics|year=2005|publisher=Wiley-VCH-Verl.|location=Weinheim|isbn=978-3527405541|edition=3rd, completely revised. and enlarged |coauthors=Trigg, George L.}}</ref> |
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:<math>\mathbf{P} = \mathbf{p} + e\mathbf{A}.</math> |
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The kinetic momentum ''m'''''v''' is not always the result this derivative. See also: [[momentum#Electromagnetism|Momentum in electromagnetism (section)]] |
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=== Hamiltonian formulation=== |
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The classical [[Hamiltonian]] ''H'' for a particle in any field equals the total energy of the system: the [[kinetic energy]] ''T'' = '''p'''<sup>2</sup>/2''m'' plus the [[potential energy]] ''V'': |
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:<math>H = T + V = \frac{\mathbf{p}^2}{2m} + V, </math> |
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where '''p'''<sup>2</sup> = '''p·p''' (see [[dot product]]). |
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For a particle in an [[electromagnetic field]], the potential energy is ''V'' = ''e''φ, and since the kinetic energy ''T'' always corresponds to the kinetic momentum '''p''', replacing the kinetic momentum by the above equation ('''p''' = '''P''' − ''e'''''A''') leads to the Hamiltonian: |
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:<math>H = \frac{(\mathbf{P} -e\mathbf{A})^2}{2m} + e\phi. </math> |
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==Canonical commutation relations== |
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The kinetic momentum ('''p''' above) satisfies the [[Commutator|commutation relation]]:<ref name=Lerner/> |
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:<math>\left [ p_j , p_k \right ] = \frac{i\hbar e}{c} \epsilon_{jk\ell } B_\ell</math> |
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where: ''j, k, l'' are indices labelling vector components, ''B<sub>l</sub>'' is a component of the [[magnetic field]], and ''ε<sub>kjl</sub>'' is the [[permutation tensor]], here in 3-dimensions. |
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==Relativistic dynamics== |
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In [[Theory of relativity|relativity]], the [[Lagrangian]] for the particle interacting with the field is |
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:<math>L = -m\sqrt{1-\left(\frac{\dot{\mathbf{r}}}{c}\right)^2} + e \mathbf{A}(\mathbf{r})\cdot\dot{\mathbf{r}} - e \phi(\mathbf{r}) \,\!</math> |
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The action is the relativistic [[arclength]] of the path of the particle in [[space time]], minus the potential energy contribution, plus an extra contribution which [[Quantum Mechanics|quantum mechanically]] is an extra [[phase (waves)|phase]] a charged particle gets when it is moving along a vector potential. |
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The momentum conjugate to '''r''', that is the canonical momentum '''P''', is defined from the variation of the lagrangian: |
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:<math> \mathbf{P} = \frac{\partial L}{\partial \dot{\mathbf{r}} } = \frac{m\dot{\mathbf{r}}}{\sqrt{1-\left(\frac{\dot{\mathbf{r}}}{c}\right)^2}} + e\mathbf{A} \,\!</math> |
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The kinetic momentum is the relativistic momentum of a particle moving with velocity ''v'' = '''ṙ''', still (''P'' – ''e'''''A'''), so we have: |
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:<math> \mathbf{P}-e\mathbf{A} = \frac{m\dot{\mathbf{r}}}{\sqrt{1-\left(\frac{\dot{\mathbf{r}}}{c}\right)^2}} \,</math> |
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The Hamiltonian equals total energy (kinetic plus potential), and is the usual relativistic expression for the energy. So in terms of the kinetic momentum: |
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:<math>H= \mathbf{P}\cdot\dot{\mathbf{r}} - L = {m\over \sqrt{1-\left(\frac{\dot{\mathbf{r}}}{c}\right)^2}} + e \phi = \sqrt{(\mathbf{P} -e\mathbf{A})^2 + (mc^2)^2} + e \phi \,</math> |
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The equations of motion derived by [[calculus of variations|extremizing]] the action (see [[matrix calculus]] for the notation): |
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:<math> \frac{\mathrm{d}\mathbf{P}}{\mathrm{d}t} =\frac{\partial L}{\partial \mathbf{r}} = e {\partial \mathbf{A} \over \partial \mathbf{r}}\cdot \dot{\mathbf{r}} - e {\partial \phi \over \partial \mathbf{r} }\,\!</math> |
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:<math>\mathbf{P} -e\mathbf{A} = \frac{m\dot{\mathbf{r}}}{\sqrt{1-\left(\frac{\dot{\mathbf{r}}}{c}\right)^2}}\,</math> |
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are the same as [[Hamiltonian mechanics|Hamilton's equations of motion]]: |
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:<math> \frac{\mathrm{d}\mathbf{r}}{\mathrm{d}t} = \frac{\partial}{\partial \mathbf{p}}\left ( \sqrt{(\mathbf{P}-e\mathbf{A})^2 +(mc^2)^2} + e\phi \right ) \,\!</math> |
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:<math> \frac{\mathrm{d}\mathbf{p}}{\mathrm{d}t} = -{\partial \over \partial \mathbf{r}}\left ( \sqrt{(\mathbf{P}-e\mathbf{A})^2 + (mc^2)^2} + e\phi \right ) \,\!</math> |
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both are equivalent to the noncanonical form: |
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:<math> \frac{\mathrm{d}}{\mathrm{d}t}\left ( {m\dot{\mathbf{r}} \over \sqrt{1-\left(\frac{\dot{\mathbf{r}}}{c}\right)^2}} \right ) = e\left ( \bold{E} + \mathbf{v} \times \mathbf{B} \right ) . \,\!</math> |
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This formula is the [[Lorentz force]], representing the rate at which the EM field adds relativistic momentum to the particle. |
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==See also== |
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*[[Momentum]] |
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*[[Canonical commutation relation]] |
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*[[Magnetic vector potential]] |
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*[[Electromagnetic field]] |
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*[[Hamiltonian]] |
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*[[Lagrangian]] |
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*[[Generalized coordinates]] |
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==References== |
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{{reflist}} |
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{{Refbegin}} |
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*{{cite book|last=Kibble|first=T.W.B.|title=Classical mechanics |edition=2d |year=1973|publisher=McGraw-Hill|location=London|isbn=9780070840188|edition=2nd}} Although concentrates on undergraduate-level classical Newtonian and Lagrangian mechanics, also contains a chapter on potential theory, includes electrodynamic fields: the ''ϕ'' and ''A'' fields and canonical momentum. |
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*{{cite book|last=Abers|first=Ernest S.|title=Quantum mechanics|year=2004|publisher=Pearson Education Inc.|location=Upper Saddle River, NJ|isbn=0-1314-6100-1}} Focuses on graduate-level quantum mechanics, but also contains similar coverage to the above. |
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*{{cite book|last=McMahon|first=David|title=Quantum field theory demystified|year=2008|publisher=McGraw-Hill|location=New York, N.Y.|isbn=978-0-07-154382-8|edition=Online-Ausg.}} Again concentrates on under/post-graduate level quantum mechanics, but does provide some exposure to Lagrangian field theory and application to the EM field. |
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{{Refend}} |
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{{DEFAULTSORT:Kinetic Momentum}} |
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[[Category:Special relativity]] |
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Revision as of 00:20, 7 September 2012
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