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.

In canonical transformations

There are four basic generating functions, summarized by the following table:

Example

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

:$H = aP^2 + bQ^2.$

For example, with the Hamiltonian

:$H = \frac + \frac,$

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

:$H = \frac + \frac,$

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,

:$P = - \frac,$

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

: $\frac = - \frac$

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

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

: $q = - \frac = \frac$

See also

*Hamilton–Jacobi equation
*Poisson bracket

References

Further reading

* {{cite book, title=Classical Mechanics, last1=Goldstein, first1=Herbert, last2=Poole, first2=C. P., last3=Safko, first3=J. L., publisher=Addison-Wesley, year=2001, isbn=978-0-201-65702-9, edition=3rd

Classical mechanics
Hamiltonian mechanics