Lagrangian and Hamiltonian Formulation of Mechanics.

The action integral for a particle of mass $m$ and charge $q$ is

$$I =\int^{\tau_2}_{\tau_1}\left\{\frac{m}{2}~\eta_{\alpha\beta} \fra... ...ac{dx^\beta}{d\tau}+ qA_\alpha {(x^\gamma)}\frac{dx^\alpha}{d\tau}\right\}d\tau$$

$$\equiv\int^{\tau_2}_{\tau_1}L(x,\dot{x})d\tau$$ (1)

where $[\eta_{\alpha\beta}]=diag(-1,1,1,1)$ and $A_\alpha$ is the given electromagnetic vector potential. The integral I is an extremum for those worldlines between $x^\alpha(\tau_1)$ and $x^\beta(\tau_2)$ which satisfy the Euler-Lagrange equation

$$\frac{d}{d\tau}~\frac{\partial L}{\partial \dot{x}^\alpha}- \frac{\partial L}{\partial x^\alpha}=0,$$

namely,

$$m\eta_{\alpha\beta}~\frac{d^2x^\beta}{d\tau^2}= q~~\frac{\partial... ...a}{d\tau}- q~\frac{\partial A_\alpha}{\partial x^\beta}~\frac{dx^\beta}{d\tau}$$

or

$$\frac{d^2x^\beta}{d\tau^2}=\frac{q}{m}~F^\beta_{~\alpha}~\frac{dx^\alpha}{d\tau} ~~$$with$$~F^\beta_{~\alpha}= \eta^{\beta \gamma}~\left(\frac{\partial A_\alpha}{\partial x^\gamma}- \frac{\partial A_\gamma}{\partial x^\alpha}\right),$$

which are the Lorentz equations of motion for the charged particle.

A physically and mathematically more advantageous set of equations is based on the introduction of the momentum variables

$$p_\alpha\equiv \frac{\partial L}{\partial\dot{x}^\alpha}=\eta_{\alpha\beta}~\frac{dx^\beta}{d\tau}m+qA_\alpha$$

and the superhamiltonian

$$\mathcal{H} =\dot{x}^\alpha~\frac{\partial L}{\partial\dot{x}^\alpha}-L$$

$$=\frac{1}{2m}~\eta^{\alpha\beta}(p_\alpha-qA_\alpha)(p_\beta-qA_\beta)~.$$ (2)

In terms of these one has the two sets of Hamiltonian equations of motion

$$\frac{dx^\alpha}{d\tau} =~~\frac{\partial\mathcal{H}}{\partial p_\alpha}= \frac{1}{m}~\eta^{\alpha\beta}(p_\beta-qA_\beta)$$ (3)
and

$$\frac{dp_\gamma}{d\tau} =-\frac{\partial\mathcal{H}}{\partial x^\gamma}= \frac{q}{m}~\eta^{\alpha\beta}(p_\alpha-qA_\alpha)~\frac{\partial A_\beta}{\partial x^\gamma}.$$ (4)

One can readily check that this system of first order equations implies the Euler-Lagrange equations.