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.

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

In canonical transformations

There are four basic generating functions, summarized by the following table:

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):

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

See also

• Hamilton–Jacobi equation
• Poisson bracket

References

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