Physics:Generating function

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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:[1]

Generating function Its derivatives
F=F1(q,Q,t) p=F1q and P=F1Q
F=F2(q,P,t)=F1+QP p=F2q and Q=F2P
F=F3(p,Q,t)=F1qp q=F3p and P=F3Q
F=F4(p,P,t)=F1qp+QP q=F4p and Q=F4P

Example

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

H=aP2+bQ2.

For example, with the Hamiltonian

H=12q2+p2q42,

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

P=pq2 and Q=1q.

 

 

 

 

(1)

This turns the Hamiltonian into

H=Q22+P22,

which is in the form of the harmonic oscillator Hamiltonian.

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

F=F3(p,Q).

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

P=F3Q,

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

pQ2=F3Q

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

F3(p,Q)=pQ

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

q=F3p=1Q

See also

References

  1. Goldstein, Herbert; Poole, C. P.; Safko, J. L. (2001). Classical Mechanics (3rd ed.). Addison-Wesley. pp. 373. ISBN 978-0-201-65702-9. 

Further reading

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