Editing Magnetic vector potential (section)

== Interpretation as Potential Momentum ==
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By equating [[Newton%27s_laws_of_motion#Second_law|Newton's second law]] with the [[Lorentz force law]] we can obtain<ref name="Semon1996" />
<math display="block">
m\frac{\mathrm{d}v}{\mathrm{d}t}=q \left(\mathbf{E} + \mathbf{v}\times\mathbf{B}   \right).</math>
Dotting this with the velocity  yields
<math display="block">
\frac{\mathrm{d}}{\mathrm{d} t}\left(\frac{1}{2}m v^2\right) = q\mathbf{v} \cdot \left(\mathbf{E} + \mathbf{v}\times\mathbf{B} \right).
</math>
With the [[dot product]] of the [[cross product]] being zero, substituting
<math display="block">\mathbf{E} =  - \nabla \phi - \frac { \partial \mathbf{A} } { \partial t },</math>
and the [[ convective derivative]] of <math>\phi</math> in the above equation then gives
<math display="block"> \frac{\mathrm{d}}{\mathrm{d} t} \left ( \frac{1}{2} mv^2 + q \phi \right ) = \frac{\partial}{\partial t} q \left ( \phi - \mathbf{v} \cdot \mathbf{A} \right ) </math>
which tells us the time derivative of the "generalized energy" <math> \frac{1}{2} mv^2 + q \phi </math> in terms of a velocity dependent potential <math> q \left ( \phi - \mathbf{v} \cdot \mathbf{A} \right ) </math>, and
<math display="block"> \frac{\mathrm{d}}{\mathrm{d} t} \left ( mv + q \mathbf{A} \right ) = - \nabla q \left ( \phi - \mathbf{v} \cdot \mathbf{A} \right ) </math>
which gives the time derivative of the [[generalized momentum]] <math> m \mathbf{v} + q \mathbf{A} </math> in terms of the (minus) gradient of the same velocity dependent potential.

Thus, when the (partial) time derivative of the velocity dependent potential <math> q (\phi - \mathbf{v} \cdot \mathbf{A}) </math> is zero, the generalized energy is conserved, and likewise when the gradient is zero, the generalized momentum is conserved. As a special case, if the potentials are time or space symmetric, then the generalized energy or momentum respectively will be conserved. Likewise the fields contribute <math> q \mathbf{r} \times \mathbf{A} </math> to the generalized angular momentum, and rotational symmetries will provide conservation laws for the components. 

Relativistically, we have the single equation
<math display="block"> \frac{\mathrm{d}}{\mathrm{d} \tau} \left ( p^{\mu} + qA^{\mu} \right ) = \partial_{\nu} \left ( U^{\mu} \cdot A^{\mu} \right ) </math>
where
* <math> \tau </math> is the [[proper time]],
* <math> p^{\mu} </math> is the [[four momentum]] <math> (E/c, \gamma m \mathbf{v}) </math>
* <math> U^{\mu} </math> is the [[four velocity]] <math> \gamma (c, \mathbf{v}) </math>
* <math> A^{\mu} </math> is the [[four potential]] <math> (\phi/c, \mathbf{A}) </math>
* <math> \partial_{\nu} </math> is the [[four gradient]] <math> (\frac{\partial}{\partial \left ( ct \right )}, -\nabla) </math>

=== Analytical Mechanics of a Charged Particle ===
In a field with electric potential <math>\ \phi\ </math> and magnetic potential <math>\ \mathbf{A}</math>, the [[Lagrangian mechanics|Lagrangian]] (<math>\ \mathcal{L}\ </math>) and the [[Hamiltonian mechanics|Hamiltonian]] (<math>\ \mathcal{H}\ </math>) of a particle with mass <math>\ m\ </math> and charge <math>\ q\ </math> are<math display="block">\begin{aligned}
\mathcal{L} &= \frac{1}{2} m\ \mathbf v^2 + q\ \mathbf  v \cdot  \mathbf A - q\ \phi\ ,\\
\mathcal{H} &= \frac{1}{2m}\left( \mathbf{p} - q\mathbf A \right)^2 + q\ \phi ~.
\end{aligned}</math>

The generalized momentum <math> \mathbf{p} </math> is <math> \frac{\partial \mathcal{L}}{\partial v} = m \mathbf{v} + q \mathbf{A} </math>. The generalized force is <math> \nabla \mathcal{L} = -q \nabla  \left ( \phi - \mathbf{v} \cdot \mathbf{A} \right ) </math>. These are exactly the quantities from the previous section. It this framework, the conservation laws come from [[Noether's theorem]].

=== Example: Solenoid ===
Consider a charged particle of charge <math> q </math> located distance <math> r </math> outside a solenoid oriented on the <math> z </math> that is suddenly turned off. By [[Faraday's law of induction]], an electric field will be induced that will impart an impulse to the particle equal to <math> q \Phi_0/2 \pi r \hat{\phi} </math> where <math> \Phi_0 </math> is the  initial [[magnetic flux]] through a cross section of the solenoid. <ref> {{cite book | last1=Feynman | first1=Richard P. | author-link1=Richard Feynman | last2=Leighton | first2=Robert B. | author-link2=Robert B. Leighton | last3=Sands | first3=Matthew | author-link3=Matthew Sands | title=The Feynman Lectures on Physics | volume=2 | chapter=17 | chapter-url=https://www.feynmanlectures.caltech.edu/II_17.html | publisher=Addison-Wesley | year=1964 | isbn=978-0-201-02115-8 }} </ref>

We can analyze this problem from the perspective of generalized momentum conservation.<ref name=Semon1996 /> Using the analogy to Ampere's law, the magnetic vector potential is <math> \mathbf{A}(r) = \Phi_0/2 \pi r \hat{\phi} </math>. Since <math> \mathbf{p} + q\mathbf{A} </math> is conserved, after the solenoid is turned off the particle will have momentum equal to <math> q \mathbf{A} = q \Phi_0/2 \pi r \hat{\phi} </math>

Additionally, because of the symmetry, the <math> z </math> component of the generalized angular momentum is conserved. By looking at the [[Poynting vector]] of the configuration, one can deduce that the fields have nonzero total angular momentum pointing along the solenoid. This is the angular momentum transferred to the fields.