The Hamilton-Jacobi Equation
Back to Configuration Space
We've established that the action, regarded as a function of its coordinate endpoints and time, satisfies
and at the same time so obeys the first-order differential equation
This is the Hamilton-Jacobi equation.
Notice that we're now back in configuration space!
For example, the Hamilton-Jacobi equation for the simple harmonic oscillator in one dimension is
(Notice that this has some resemblance to the Schrödinger equation for the same system.)
If the Hamiltonian has no explicit time dependence becomes just
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