Physics:Generating function

Short description: Function used to generate other functions

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:

Generating function	Its derivatives
[math]\displaystyle{ F= F_1(q, Q, t) \,\! }[/math]	[math]\displaystyle{ p = ~~\frac{\partial F_1}{\partial q} \,\! }[/math] and [math]\displaystyle{ P = - \frac{\partial F_1}{\partial Q} \,\! }[/math]
[math]\displaystyle{ F= F_2(q, P, t) = F_1 + QP \,\! }[/math]	[math]\displaystyle{ p = ~~\frac{\partial F_2}{\partial q} \,\! }[/math] and [math]\displaystyle{ Q = ~~\frac{\partial F_2}{\partial P} \,\! }[/math]
[math]\displaystyle{ F= F_3(p, Q, t) =F_1 - qp \,\! }[/math]	[math]\displaystyle{ q = - \frac{\partial F_3}{\partial p} \,\! }[/math] and [math]\displaystyle{ P = - \frac{\partial F_3}{\partial Q} \,\! }[/math]
[math]\displaystyle{ F= F_4(p, P, t) =F_1 - qp + QP \,\! }[/math]	[math]\displaystyle{ q = - \frac{\partial F_4}{\partial p} \,\! }[/math] and [math]\displaystyle{ Q = ~~\frac{\partial F_4}{\partial P} \,\! }[/math]

Example

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

[math]\displaystyle{ H = aP^2 + bQ^2. }[/math]

For example, with the Hamiltonian

[math]\displaystyle{ H = \frac{1}{2q^2} + \frac{p^2 q^4}{2}, }[/math]

where p is the generalized momentum and q is the generalized coordinate, a good canonical transformation to choose would be

[math]\displaystyle{ P = pq^2 \text{ and }Q = \frac{-1}{q}. \, }[/math]

(1)

This turns the Hamiltonian into

[math]\displaystyle{ H = \frac{Q^2}{2} + \frac{P^2}{2}, }[/math]

which is in the form of the harmonic oscillator Hamiltonian.

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

[math]\displaystyle{ F = F_3(p,Q). }[/math]

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

[math]\displaystyle{ P = - \frac{\partial F_3}{\partial Q}, }[/math]

and substitute the expression for P from equation (1), expressed in terms of p and Q:

[math]\displaystyle{ \frac{p}{Q^2} = - \frac{\partial F_3}{\partial Q} }[/math]

Integrating this with respect to Q results in an equation for the generating function of the transformation given by equation (1):

[math]\displaystyle{ F_3(p,Q) = \frac{p}{Q} }[/math]

To confirm that this is the correct generating function, verify that it matches (1):

[math]\displaystyle{ q = - \frac{\partial F_3}{\partial p} = \frac{-1}{Q} }[/math]

See also

• Hamilton–Jacobi equation
• Poisson bracket

Further reading

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