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
F= F_1(q, Q, t) \,\! p = ~~\frac{\partial F_1}{\partial q} \,\! and P = - \frac{\partial F_1}{\partial Q} \,\!
F= F_2(q, P, t) = F_1 - QP \,\! p = ~~\frac{\partial F_2}{\partial q} \,\! and Q = ~~\frac{\partial F_2}{\partial P} \,\!
F= F_3(p, Q, t) =F_1 + qp \,\! q = - \frac{\partial F_3}{\partial p} \,\! and P = - \frac{\partial F_3}{\partial Q} \,\!
F= F_4(p, P, t) =F_1 + qp - QP \,\! q = - \frac{\partial F_4}{\partial p} \,\! and Q = ~~\frac{\partial F_4}{\partial P} \,\!

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{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

P = pq^2 \text{ and }Q = \frac{-1}{q}. \,

 

 

 

 

(1)

This turns the Hamiltonian into

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,

F = F_3(p,Q).

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

P = - \frac{\partial F_3}{\partial Q},

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

\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):

F_3(p,Q) = \frac{p}{Q}

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

q = - \frac{\partial F_3}{\partial p} = \frac{-1}{Q}

See also

References


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