# Generating function (physics)

**Generating functions** which arise in Hamiltonian mechanics are quite different from generating functions in mathematics. In physics, 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 |
---|---|

and | |

and | |

and | |

and |

## Example

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

For example, with the Hamiltonian

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

which is in the form of the harmonic oscillator Hamiltonian.

The generating function *F* for this transformation is of the third kind,

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

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

Integrating this with respect to *Q* results in an equation for the generating function of the transformation given by equation (Template:EquationNote):

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

## See also

## References

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