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]

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

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

For example, with the Hamiltonian

$${\displaystyle H={\frac {1}{2q^{2}}}+{\frac {p^{2}q^{4}}{2}},}$${\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

This turns the Hamiltonian into

$${\displaystyle H={\frac {Q^{2}}{2}}+{\frac {P^{2}}{2}},}$${\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,

$${\displaystyle F=F_{3}(p,Q).}$${\displaystyle F=F_{3}(p,Q).}

To find F explicitly, use the equation for its derivative from the table above,

$${\displaystyle P=-{\frac {\partial F_{3}}{\partial Q}},}$${\displaystyle P=-{\frac {\partial F_{3}}{\partial Q}},}

and substitute the expression for P from equation (1 ), expressed in terms of p and Q:

$${\displaystyle {\frac {p}{Q^{2}}}=-{\frac {\partial F_{3}}{\partial Q}}}$${\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 ):

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

• Hamilton–Jacobi equation
• Poisson bracket

• Classical mechanics
• Hamiltonian mechanics

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• Short description is different from Wikidata