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:^[1]

Generating function	Its derivatives
${\displaystyle F=F_{1}(q,Q,t)\,\!}$	${\displaystyle p=~~{\frac {\partial F_{1)){\partial q))\,\!}$ and ${\displaystyle P=-{\frac {\partial F_{1)){\partial Q))\,\!}$
${\displaystyle F=F_{2}(q,P,t)=F_{1}+QP\,\!}$	${\displaystyle p=~~{\frac {\partial F_{2)){\partial q))\,\!}$ and ${\displaystyle Q=~~{\frac {\partial F_{2)){\partial P))\,\!}$
${\displaystyle F=F_{3}(p,Q,t)=F_{1}-qp\,\!}$	${\displaystyle q=-{\frac {\partial F_{3)){\partial p))\,\!}$ and ${\displaystyle P=-{\frac {\partial F_{3)){\partial Q))\,\!}$
${\displaystyle F=F_{4}(p,P,t)=F_{1}-qp+QP\,\!}$	${\displaystyle q=-{\frac {\partial F_{4)){\partial p))\,\!}$ and ${\displaystyle Q=~~{\frac {\partial F_{4)){\partial P))\,\!}$

Example

Sometimes a given Hamiltonian can be turned into one that looks like the harmonic oscillator Hamiltonian, which is

${\displaystyle H=aP^{2}+bQ^{2}.}$

For example, with the Hamiltonian

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

where p is the generalized momentum and q is the generalized coordinate, a good canonical transformation to choose would be

${\displaystyle P=pq^{2}{\text{ and ))Q={\frac {-1}{q)).\,}$

(1)

This turns the Hamiltonian into

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

which is in the form of the harmonic oscillator Hamiltonian.

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

${\displaystyle F=F_{3}(p,Q).}$

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

${\displaystyle P=-{\frac {\partial F_{3)){\partial Q)),}$

and substitute the expression for P from equation (1), expressed in terms of p and Q:

${\displaystyle {\frac {p}{Q^{2))}=-{\frac {\partial F_{3)){\partial Q))}$

Integrating this with respect to Q results in an equation for the generating function of the transformation given by equation (1):

${\displaystyle F_{3}(p,Q)={\frac {p}{Q))}$

To confirm that this is the correct generating function, verify that it matches (1):

${\displaystyle q=-{\frac {\partial F_{3)){\partial p))={\frac {-1}{Q))}$