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
(*q*_{i}, *p*_{i}) and *H* satisfy Hamilton equations then
the new
(*Q*_{i}, *P*_{i}) 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
a function

called a

p_{i} |
= | (4.26) | |

Q_{i} |
= | (4.27) | |

= | (4.28) |

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:

therefore

If

and

This is the Hamilton-Jacobi equation.

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

where and

Here is how this comes about.

Start from the Schrödinger equation for a single quantum
``particle'':

(4.31) |

where , and

There is a little more work that we have to do with the Laplacian on the right hand side of the Schrödinger equation. Let us evaluate first:

(4.33) |

Acting with again yields:

Now we have to substitute equations (4.36) and (4.38) into the Schrödinger equation, divide both sides by , this term will accompany all other terms, and collect separately real and imaginary parts of the resulting equation.

Collecting the real part yields:

(4.35) |

Dividing this equation by and grouping all terms where cancels out on the left hand side yields:

The imaginary part, in turn, yields

(4.37) |

Multiplying both sides of this equation by lets us rewrite it finally as:

If you look at the real part of the Schrödinger equation, equation (4.40), the one that gives , you can see that in the limit it turns into:

(4.39) |

which is the Hamilton-Jacobi equation for a classical particle moving in a potential field

(4.40) |

is called a

But returning to the Hamilton-Jacobi equation, in summary, what we have just demonstrated is that:

The Hamilton-Jacobi functionSis, on the one hand, the forming function of a canonical transformation that annihilates the Hamiltonian, and, on the other, it is thephaseof the quantum mechanical wave function that represents a quantum particle.