The Hamilton-Jacobi Equations for a Relativistic Charged Particle

In the problem of finding the motion of a classical particle one has the choice of dealing with a system of second order ordinary differential equations (Lagrange's equations) or a single first order partial differential equation (the Hamilton-Jacobi equation, henceforth referred to as the H-J equation). In practice the latter method is less frequently used because of the difficulty in finding complete integrals. When these are obtainable, however, the method leads rapidly to the equations of the trajectories. Furthermore it is of fundamental theoretical importance and it provides a basis for quantum mechanical analogues.