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


 * $$H = aP^2 + bQ^2.$$

For example, with the Hamiltonian


 * $$H = \frac{1}{2q^2} + \frac{p^2 q^4}{2},$$

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


 * $$H = \frac{Q^2}{2} + \frac{P^2}{2},$$

which is in the form of the harmonic oscillator Hamiltonian.

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


 * $$F = F_3(p,Q).$$

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


 * $$P = - \frac{\partial F_3}{\partial Q},$$

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


 * $$\frac{p}{Q^2} = - \frac{\partial F_3}{\partial Q}$$

Integrating this with respect to Q results in an equation for the generating function of the transformation given by equation ($$):
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 * $$F_3(p,Q) = \frac{p}{Q}$$
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To confirm that this is the correct generating function, verify that it matches ($$):


 * $$q = - \frac{\partial F_3}{\partial p} = \frac{-1}{Q}$$