In canonical transformations
There are four basic generating functions, summarized by the following table: [1]
| Generating function | Its derivatives |
|---|---|
| and | |
| and | |
| and | |
| and |
In physics, and more specifically in Hamiltonian mechanics, 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.
There are four basic generating functions, summarized by the following table: [1]
| Generating function | Its derivatives |
|---|---|
| and | |
| and | |
| and | |
| and |
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
| (1) |
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 ):
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