Hamilton–Jacobi equation

In the Hamilton -Jacobi formalism (named after the mathematicians William Rowan Hamilton and Carl Gustav Jacob Jacobi ) is a special canonical transformation

A Hamiltonian generated which is equal to zero for all times.

This has the consequence that both the transformed spatial coordinates, as well as their canonically conjugate momenta are conserved quantities, so that all dynamic variables are cyclic coordinates in the new Hamiltonian function:

The transformed Hamiltonian function is obtained by adding the partial time derivative of a generating function to the untransformed Hamiltonian.

It is specifically chosen a generating function, taken from the location coordinates and the new (constant) pulses depends so

Used in results in the Hamilton -Jacobi differential equation for:

It is a partial differential equation in the variables and for the Hamiltonian action function (the use of the term "effect" is justified below).

The transformed equations of motion are trivial, but the problem is shifted to finding a matching generatrix.

Hamilton-Jacobi formalism for not explicitly time-dependent Hamiltonian

The aim of the Hamilton -Jacobi formalism is to simplify the Hamiltonian equations of motion by means of a canonical transformation. To a generating function to the initial Hamiltonian function, which depends on the old and pulses places constructed that it transforms into a new Hamiltonian function, which depends only on the new (constant) pulse

For conservative systems (not explicitly time-dependent) are the new momentum constants of the motion. The new locations only change linearly with time

Must apply for

Inducted into the Hamiltonian function results

This is the Hamilton-Jacobi equation for conservative systems. They determined.

As an illustration of the total time derivative is calculated

To use the Lagrange's equations of motion now (with the Lagrangian, wherein the kinetic energy is, the potential ):

Provides the temporal integration

Is thus the same as the effect of integral.

Example: The one-dimensional harmonic oscillator

Be an arbitrary potential. The Hamiltonian function is

The Hamilton-Jacobi equation

In the one-dimensional oscillator is the only constant of the motion. There must also be constant, put what is possible for all conservative systems.

By integrating follows

With

Because of Hamilton 's equation of motion also applies

In order to show the movement in and must be transformed back to the old coordinates

For the special case of the harmonic oscillator is obtained with

Thus

And ultimately