Generating functions which arise in Hamiltonian mechanics are quite different from generating functions in mathematics. In physics, a generating function is, loosely, a function whose partial derivatives generate the differential equations that determine a system's dynamics. Common examples are the partition function of statistical mechanics, the Hamiltonian, and the function which acts as a bridge between two sets of canonical variables when performing a canonical transformation.
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In Canonical Transformations
There are four basic generating functions, summarized by the following table:
Example
Sometimes a given Hamiltonian can be turned into one that looks like the harmonic oscillator Hamiltonian, which is
For example, with the Hamiltonian
where p is the 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 third kind,
To find F explicitly, use the equation for its derivative from the table above,
and substitute the expression for P from equation (1), 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 (1):
To confirm that this is the correct generating function, verify that it matches (1):