Function used to generate other functions
This article is about generating functions in physics. For generating functions in mathematics, see
Generating function.
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:
[1]
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
-
|
|
(1)
|
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 (
1), 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 (
1):
|
To confirm that this is the correct generating function, verify that it matches (
1):
See also
References
-
^ Goldstein, Herbert; Poole, C. P.; Safko, J. L. (2001). Classical Mechanics (3rd ed.). Addison-Wesley. p. 373.
ISBN
978-0-201-65702-9.
Further reading