Generating function (physics)

This article is about generating functions in physics. For generating functions in mathematics, see Generating function.

Generating a sine from a circle.

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:

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

Example

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

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

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

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

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 (1), 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 (1):

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

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

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

See also

• Hamilton–Jacobi equation
• Poisson bracket