Generating function (physics)

Generating functions which arise in Hamiltonian mechanics are quite different from generating functions in mathematics. In physics, 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 $P=-{\frac {\partial F_{1}}{\partial Q}}\,\!$ $F=F_{2}(q,P,t)=F_{1}+QP\,\!$ $p=~~{\frac {\partial F_{2}}{\partial q}}\,\!$ and $Q=~~{\frac {\partial F_{2}}{\partial P}}\,\!$ $F=F_{3}(p,Q,t)=F_{1}-qp\,\!$ $q=-{\frac {\partial F_{3}}{\partial p}}\,\!$ and $P=-{\frac {\partial F_{3}}{\partial Q}}\,\!$ $F=F_{4}(p,P,t)=F_{1}-qp+QP\,\!$ $q=-{\frac {\partial F_{4}}{\partial p}}\,\!$ 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):

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