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

${\displaystyle H=aP^{2}+bQ^{2}.}$ 

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):

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

• Hamilton–Jacobi equation
• Poisson bracket

• Goldstein, Herbert; Poole, C. P.; Safko, J. L. (2001). Classical Mechanics (3rd ed.). Addison-Wesley. ISBN 978-0-201-65702-9.