Show equivalence of Force to Lorentz Force

[vinceclortho]

Greetings,

I am trying to to show that the force in this form:

$\dot{P}=e(\vec{∇}\vec{A})\vec{\dot{x}}-e\vec{∇}\phi$

Is equal to the lorentz force. I have been trying the approach of some sort of vector identity but have not gotten anywhere.

The equation and where I am trying to get with it are posted on wikipedia towards the bottom:
http://en.wikipedia.org/wiki/Hamiltonian_mechanics

I appreciate any help!

 

[vanhees71]

The equation of motion is given by (in Heaviside-Lorentz units)
$$m \frac{\mathrm{d}}{\mathrm{d} t} \left (\frac{\dot{\vec{x}}}{\sqrt{1-\dot{\vec{x}}/c^2}} \right)=q \left (\vec{E}+ \frac{\vec{v}}{c} \times \vec{B} \right )=q \left [-\vec{\nabla} \phi + \frac{\vec{v}}{c} \times (\vec{\nabla} \times \vec{A}) \right ].$$
To derive it from the principle of least action, I prefer the Lagrangian form, because there the action is manifestly covariant:
$$L=-mc^2 \sqrt{1-\vec{v}^2/c^2}-q \left (\phi-\frac{\dot{\vec{x}}}{c} \cdot \vec{A} \right ).$$
The action is manifestly covariant, because
$$\sqrt{1-v^2/c^2} \mathrm{d} t = \mathrm{d} \tau = \sqrt{\mathrm{d} x_{\mu} \mathrm{d} x^{\mu}}$$
and
$$(\phi-\frac{\dot{\vec{x}}}{c} \cdot \vec{A}) \mathrm{d} t = \mathrm{d} t \frac{1}{c} \frac{\mathrm{d} x^{\mu}}{\mathrm{d} t} A_{\mu}=\frac{1}{c} \mathrm{d} x^{\mu} A_{\mu}.$$
The four potential $(A^{\mu})=(\phi,\vec{A})$ is a Mikowski four-vector.

To get the equations of motion, you just write down the Euler-Lagrange equations
$$\frac{\mathrm{d}}{\mathrm{d} t} \frac{\partial L}{\partial \dot{\vec{x}}}=\frac{\partial L}{\partial \vec{x}}.$$
Just taking the derivatives, gives
$$\vec{P}=\frac{\partial L}{\partial \dot{\vec{x}}}=\frac{m \dot{\vec{x}}}{\sqrt{1-\dot{\vec{x}}^2/c^2}}+\frac{q}{c} \vec{A}.$$
Here it is important to note that $\vec{P}$ is the canonical momentum, not the mechanical momentum, which is given by the first term only.

Further we have
$$\frac{\partial L}{\partial \vec{x}}=-q \vec{\nabla} \phi + \frac{q}{c} \vec{\nabla} (\dot{\vec{x}} \cdot \vec{A}).$$

Written in the three-dimensional Ricci calculus (with all indices as subscripts) the equation of motion thus reads
$$\dot{P}_j=\frac{\mathrm{d}}{\mathrm{d} t} \left (\frac{m \dot{x}_j}{\sqrt{1-\dot{\vec{x}}^2/c^2}} + q A_j \right ) = \frac{\mathrm{d}}{\mathrm{d} t} \left (\frac{m \dot{x}_j}{\sqrt{1-\dot{\vec{x}}^2/c^2}} \right ) + \frac{q}{c}\dot{x}_k \partial_k A_j = -q \partial_j \phi + \frac{q}{c} \dot{x}_k \partial_j A_k.$$
Combining the two terms with the vector potential on the right-hand side gives
$$\frac{\mathrm{d}}{\mathrm{d} t} \left (\frac{m \dot{x}_j}{\sqrt{1-\dot{\vec{x}}^2/c^2}} \right )=-q \partial_j \phi + \frac{q}{c} \dot{x}_k (\partial_j A_k-\partial_k A_j).$$
The last term can be rewritten as
$$\dot{x}_k (\partial_j A_k-\partial_k A_j) =\dot{x}_k \epsilon_{jkl} (\vec{\nabla} \times \vec{A})_l=[\dot{\vec{x}} \times (\vec{\nabla} \times \vec{A})]_j.$$
Rewriting everything in vector notation yields the correct equation of motion.

Note that the notation in Wikipedia, from where you copied your equation, is not very clear. That's why I used the Ricci calculus in the intermediate steps, where it is clear what are free indices and which are repeated indices to be summed.

 

[Meir Achuz]

$${\bf F}=-e\nabla\phi-e\partial_t{\bf A}+e{\bf v\times(\nabla\times A)}$$
$$=-e\nabla\phi-e\partial_t{\bf A}+e\nabla{\bf(v\cdot A)}-e{\bf(v\cdot\nabla)A}$$
$$=-e\nabla\phi+e\nabla{\bf(v\cdot A)}-e\frac{\bf dA}{dt}.$$