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.

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

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):

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

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

• Hamilton–Jacobi equation
• Poisson bracket