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 $P = - \frac{\partial F_1}{\partial Q} \,\!$
${\displaystyle F=F_{2}(q,P,t)=F_{1}+QP\,\!}$	$p = ~~\frac{\partial F_2}{\partial q} \,\!$ and $Q = ~~\frac{\partial F_2}{\partial P} \,\!$
${\displaystyle F=F_{3}(p,Q,t)=F_{1}-qp\,\!}$	$q = - \frac{\partial F_3}{\partial p} \,\!$ and $P = - \frac{\partial F_3}{\partial Q} \,\!$
${\displaystyle F=F_{4}(p,P,t)=F_{1}-qp+QP\,\!}$	$q = - \frac{\partial F_4}{\partial p} \,\!$ 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):

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

See also

• Hamilton–Jacobi equation
• Poisson bracket

References

• Goldstein, Herbert (2002). Classical Mechanics. Addison Wesley. ISBN 978-0-201-65702-9.