Schur orthogonality relations

Generating functions which arise in Hamiltonian mechanics are quite different from generating functions in mathematics. In physics, a generating function acts as a bridge between two sets of canonical variables when performing a canonical transformation.

Details

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

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

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

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 (Template:EquationNote), 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 (Template:EquationNote):

$F_3(p,Q)=\frac{p}{Q}$

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

References

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