Generating functions which arise in Hamiltonian mechanics

Hamiltonian mechanics

Hamiltonian mechanics is a reformulation of classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton.It arose from Lagrangian mechanics, a previous reformulation of classical mechanics introduced by Joseph Louis Lagrange in 1788, but can be formulated without...

 are quite different from generating functions in mathematics. In the case of physics, generating functions act as a bridge between two sets of canonical variables when performing canonical transformation

Canonical transformation

In Hamiltonian mechanics, a canonical transformation is a change of canonical coordinates  →  that preserves the form of Hamilton's equations , although it...

.

Details

There are four basic generating functions, summarized by the following table.

Generating Function	Its Derivatives
	and
	and
	and
	and

Example

Sometimes one can turn a given Hamiltonian into one that looks a bit more like the harmonic oscillator

Harmonic oscillator

In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force, F, proportional to the displacement, x: \vec F = -k \vec x \, where k is a positive constant....

Hamiltonian, which is

So, as an example, if one were given the Hamiltonian


(where p is generalized momentum, and q is the generalized coordinate.)

a good canonical transformation to choose would be

This turns the Hamiltonian into


which is in the form of the harmonic oscillator Hamiltonian.

The generating function, F, for this transformation is of the 3rd kind,

To find F explicitly, use the equation for its derivative (from the table above),


and substitute the expression for P from equation (2), expressed in terms of p and Q:

Integrating this with respect to Q results in an equation for the generating function of the transformation given by equation (2):

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To confirm that this is the correct generating function, verify that it matches (2):