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

Let and be two worldlines having the same starting point

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

and |

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

The fact that satisfies Lagrange's equation of motion implies that the integral vanishes. Recalling the definition of , or looking at Figure 4, one sees that at the two termination points one has

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

Here

are the momentum components and

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

Thus the differential equation for the dynamical phase function is

or explicitly

This is the Hamilton-Jacobi for a charged particle in an electromagnetic vector potential .