Generating function (physics) explained

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
F = F _{1(q,} Q, t)	p = ~~ (\partial F _{1})/(\partial q) and P = - (\partial F _{1})/(\partial Q) \
F = F _{2(q,} P, t) = F _{1} + QP	p = ~~ (\partial F _{2})/(\partial q) and Q = ~~ (\partial F _{2})/(\partial P) \
F = F _{3(p,} Q, t) = F _{1} - qp	q = - (\partial F _{3})/(\partial p) and P = - (\partial F _{3})/(\partial Q) \
F = F _{4(p,} P, t) = F _{1} - qp + QP	q = - (\partial F _{4})/(\partial p) and Q = ~~ (\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 = (1)/(2 q ^{2}) + (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

This turns the Hamiltonian into

H = (Q ^{2})/(2) + (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 = - (\partial F _{3})/(\partial Q) ,

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

(p)/(Q ^{2}) = - (\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 :

F _{3(p,Q)} = (p)/(Q)

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

q = - (\partial F _{3})/(\partial p) = (-1)/(Q)

See also

• Hamilton–Jacobi equation
• Poisson bracket

Further reading

• Book: Herbert Goldstein. Charles P. Poole. Classical Mechanics. Goldstein. Herbert. Poole. C. P.. Safko. J. L.. Addison-Wesley. 2001. 978-0-201-65702-9. 3rd.

Notes and References

1. Book: Goldstein. Herbert. Classical Mechanics. Poole. C. P.. Safko. J. L.. Addison-Wesley. 2001. 978-0-201-65702-9. 3rd. 373.