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 :