Generating function (physics)

Generating a sine from a circle.

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.

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 $P=-\frac{\partial F_1}{\partial Q}$
$\begin{array}{cc}F& =F_2(q,P,t)\\ & =F_1+QP\end{array}$	$p=\frac{\partial F_2}{\partial q}$ and $Q=\frac{\partial F_2}{\partial P}$
$\begin{array}{cc}F& =F_3(p,Q,t)\\ & =F_1-qp\end{array}$	$q=-\frac{\partial F_3}{\partial p}$ and $P=-\frac{\partial F_3}{\partial Q}$
$\begin{array}{cc}F& =F_4(p,P,t)\\ & =F_1-qp+QP\end{array}$	$q=-\frac{\partial F_4}{\partial p}$ 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

$$H=a{P}^2+b{Q}^2.$$

For example, with the Hamiltonian

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

where Template:Mvar is the generalized momentum and Template:Mvar is the generalized coordinate, a good canonical transformation to choose would be

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 Template:Math for this transformation is of the third kind,

$$F=F_3(p,Q).$$

To find Template:Math explicitly, use the equation for its derivative from the table above,

$$P=-\frac{\partial F_3}{\partial Q},$$

and substitute the expression for Template:Mvar from equation (Template:EquationNote), expressed in terms of Template:Mvar and Template:Mvar:

$$\frac{p}{Q^2}=-\frac{\partial F_3}{\partial Q}$$

Integrating this with respect to Template:Mvar results in an equation for the generating function of the transformation given by equation (Template:EquationNote): Template:Equation box 1

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

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

See also

• Hamilton–Jacobi equation
• Poisson bracket