There is nothing stopping us from transforming variables on the phase space from to some new . Normally, if you change variables this way the functional form of a Hamiltonian is going to change too. In general it is not the case that if (qi, pi) and H satisfy Hamilton equations then the new (Qi, Pi) and will satisfy Hamilton equations too.
But it turns out that if the transformation
satisfies certain conditions then Hamilton equations
are preserved. Such transformations are called canonical
transformations and the condition, it turns out, is as follows:
for a transformation to be canonical
there must exist
This observation provides us with the means to solve
Hamilton equations and find quite easily all constants of motion.
Imagine that we have found a canonical transformation,
given by some forming function S, such that the new Hamiltonian is
zero. Then from Hamilton equations it follows that:
The Hamilton-Jacobi equation is enormously useful in solving analytically and numerically equations of motion for classical particles. The main reason for its usefulness is that it yields all constants of motion automatically, and the solution itself becomes formulated in terms of those constants of motion.
Another interesting feature of this equation is that the forming function S behaves a little like a wave. It can be shown that particle trajectories pierce surfaces of constant S.
A yet another interesting feature of the Hamilton-Jacobi equation
is that it can be easily derived from the Schrödinger equation
of Quantum Mechanics by representing the wave function
in the polar form
Here is how this comes about.
Start from the Schrödinger equation for a single quantum
Collecting the real part yields:
But returning to the Hamilton-Jacobi equation, in summary, what we have just demonstrated is that:
The Hamilton-Jacobi function S is, on the one hand, the forming function of a canonical transformation that annihilates the Hamiltonian, and, on the other, it is the phase of the quantum mechanical wave function that represents a quantum particle.