Solution to the Hamilton-Jacobi Equation

As an example, consider the problem of solving the H-J equation for a charged particle in the vector potential

$$\{A_0, A_1, A_2, A_3\}=\{0, A_x(t-z), A_y(t-z), 0\}.$$ (12)

It expresses a generic plane wave traveling towards positive $z$. The H-J equation is

$${ \frac{1}{2m}\left\{ -\left(\frac{\partial S}{\partial t}\right)^2 +\left(\frac{\partial S}{\partial z}\right)^2+\right.}~$$

$$\left. \left(\frac{\partial S}{\partial x}-qA_x(t-z)\right)^2 +\l... ...}{\partial y}-qA_y(t-z)\right)^2 \right\} +\frac{\partial S}{\partial \tau}=0~.$$

To make this equation physically more transparent and mathematically more manageable one introduces the retarded and advanced time coordinates

$$t-z\equiv u$$

and

$$t+z\equiv v$$

In terms of these the H-J equation becomes

$$\frac{1}{2m}\left\{ -4~\frac{\partial S}{\partial u} ~ \frac{\par... ...ial S}{\partial y}-qA_y(u)\right)^2 \right\} +\frac{\partial S}{\partial\tau}=0$$ (13)

This equation is readily solved by the method of separation of variables according to which the solution,

$$S=\int^u p_u (u) du+\int^v p_v (v)dv+\int^x p_x (x) dx+\int^y p_y (y)dy\, +\int^\tau p_\tau(\tau) d\tau~,$$

is a sum of antiderivatives, each one depending only on its respective integration variable. With this stipulation one finds that the components of the gradient of $S$ are

$$\frac{\partial S}{\partial u} \equiv p_u = \frac{1}{4P_v}\left[(P_x-qA_x(u))^2+(P_y-qA_y(u) )^2\right] +2mP_\tau$$ (14)

$$\frac{\partial S}{\partial v} \equiv p_v=P_v$$ (15)

$$\frac{\partial S}{\partial x} \equiv p_x=P_x$$ (16)

$$\frac{\partial S}{\partial y} \equiv p_y=P_y$$ (17)

$$\frac{\partial S}{\partial\tau} \equiv p_\tau=P_\tau$$ (18)

where $P_v$, $P_x$, $P_y$, as well as $P_\tau$, are the constants of separation¹. Consequently, the Hamilton-Jacobi function (``Hamilton's principal function'', ``Schroedinger phase'', dynamical phase)

$$S=S(u,v,x,y; P_\tau, P_v, P_x, P_y;\tau)$$ (19)

has the form

$$S= \frac{1}{4P_v}\int_{u_0}^u\left[(P_x-qA_x)^2+(P_y-qA_y)^2 +2mP_\tau\right]du +vP_v$$

$$+xP_x+yP_y +\tau P_\tau$$ (20)

or more generally

$$S'=S-\beta(P_\tau, P_v, P_x, P_y)~.$$

Both $S$ and $S'$ are solutions to the H-J equation, but $S'$ has an additive constant $\beta(P_\tau, P_v, P_x, P_y)$ which is a function of the four separation constants. Although the difference between $S$ and $S'$ is trivial from the perspective of solving the H-J equation, the opposite is true from the viewpoint of physics and mathematics. Indeed, suppose one let's

$$\beta=S(u_0,v_0,x_0,y_0; P_\tau, P_v, P_x, P_y;\tau_0)~.$$

Then one obtains

$$S'= \frac{1}{4P_v}\int_{u_0}^u\left[(P_x-qA_x)^2+(P_y-qA_y)^2 +\frac{mP_\tau}{2P_v}\right]du +(v-v_0)P_v$$

$$+(x-x_0)P_x+(y-y_0)P_y+(\tau -\tau_0)P_\tau~.$$ (21)

The usefulness of this H-J function $S'$ is that it generates new phase space coordinates relative to which the solutions of Hamilton's Eqs.(3)-(4) are straight lines. They are depicted in Figure 8 on page . In other words, solving the H-J equation for $S'$ is tantamount to solving Hamilton's equations for all possible initial conditions.