Generating function (physics)

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