The Hamilton-Jacobi Equation for a Relativistic Particle.

Being defined in terms of the action integral, the dynamical phase satisfies a differential equation which one obtains by a simple argument:

Let $x^\alpha(\tau)$ and $\overline{x}^\alpha(\tau)=x^\alpha(\tau)+h^\alpha(\tau)$ be two worldlines having the same starting point

$$x^\alpha(\tau_1)=\overline{x}^\alpha(\tau_1),$$

both satisfying Lagrange's equation of motion, but having slightly different termination points

$$x^\alpha(\tau) ~ ~ \overline{x}^\alpha(\tau+\delta\tau)= x^\alpha(\tau)+h^\alpha(\tau)+\dot{x}^\alpha\delta\tau+\cdots$$

as in Figure 4. Then the (principal linear part of the) difference in the value of the dynamical phase at these termination points is

$$\delta S = \int\limits^{\tau+\delta\tau}_{\tau_1}L(\overline{x}^\alpha,\do... ...rline{x}}^\alpha)d\tau-\int\limits^\tau_{\tau_1}L(x^\alpha,\dot{x}^\alpha)d\tau$$

$$=\int\limits^\tau_{\tau_1}\sum\limits_\alpha\left( \frac{\partial... ...t^{\tau+\delta\tau}_{\tau_1}+L\delta\tau\bigg\vert^{\tau+\delta\tau}_{\tau_1}~.$$

Figure 4: Differential of the action function $S$ as a function of variations $\delta \tau$ and $\delta x^\alpha$ in the endpoint of an extremal world line.

The fact that $x^\alpha(\tau)$ satisfies Lagrange's equation of motion implies that the integral vanishes. Recalling the definition of $\delta x^\alpha$, or looking at Figure 4, one sees that at the two termination points one has

$$h^\alpha=\delta x^\alpha-\dot{x}\delta\tau.$$

Consequently, the principal linear part of the difference between the two $S$ values at the termination point is

$$\delta S=\sum\limits_\alpha p_\alpha\delta x^\alpha-\mathcal{H}\delta\tau.$$ (10)

Here

$$p_\alpha=\frac{\partial L}{\partial\dot{x}^\alpha}\left(=\eta_{\alpha\beta}\dot{x}^\beta+qA_\alpha\right)$$

are the momentum components and

$$\mathcal{H}\equiv\frac{\partial L}{\partial\dot{x}^\alpha}~\dot{x}^\alpha-L =\frac{\eta^{\alpha\beta}}{2m}(p_\alpha-qA_\alpha)(p_\beta-qA_\beta)$$

is the superhamiltonian of the charged particle at the termination point of its worldline. Equation (11) is the expression for the differential of $S$. One has

$$\frac{\partial S}{\partial x^\alpha} =p_\alpha$$

$$\frac{\partial S}{\partial\tau} =-\mathcal{H}(x^\alpha,p_\gamma)$$

Thus the differential equation for the dynamical phase function $S$ is

$$\mathcal{H}\left(x^\alpha,\frac{\partial S}{\partial x^\gamma}\right)+\frac{\partial S}{\partial\tau}=0,$$ (11)

or explicitly

$$\frac{\eta^{\alpha\beta}}{2m}\left( \frac{\partial S}{\partial x^... ...rtial S}{\partial x^\beta}-qA_\beta\right) +\frac{\partial S}{\partial \tau}=0.$$

This is the Hamilton-Jacobi for a charged particle in an electromagnetic vector potential $A_\alpha(x)$.