# Generating function (physics)

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Function used to generate other functions

This article is about generating functions in physics. For generating functions in mathematics, see [Generating function](/source/Generating_function).

 Generating a sine from a circle.

In physics, and more specifically in [Hamiltonian mechanics](/source/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](/source/Partition_function_(statistical_mechanics)) of [statistical mechanics](/source/Statistical_mechanics), the Hamiltonian, and the function which acts as a bridge between two sets of canonical variables when performing a [canonical transformation](/source/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 ) {\displaystyle F=F_{1}(q,Q,t)} p = ∂ F 1 ∂ q {\displaystyle p=~~{\frac {\partial F_{1}}{\partial q}}\,\!} and P = − ∂ F 1 ∂ Q {\displaystyle P=-{\frac {\partial F_{1}}{\partial Q}}\,\!} F = F 2 ( q , P , t ) = F 1 + Q P {\displaystyle {\begin{aligned}F&=F_{2}(q,P,t)\\&=F_{1}+QP\end{aligned}}} p = ∂ F 2 ∂ q {\displaystyle p=~~{\frac {\partial F_{2}}{\partial q}}\,\!} and Q = ∂ F 2 ∂ P {\displaystyle Q=~~{\frac {\partial F_{2}}{\partial P}}\,\!} F = F 3 ( p , Q , t ) = F 1 − q p {\displaystyle {\begin{aligned}F&=F_{3}(p,Q,t)\\&=F_{1}-qp\end{aligned}}} q = − ∂ F 3 ∂ p {\displaystyle q=-{\frac {\partial F_{3}}{\partial p}}\,\!} and P = − ∂ F 3 ∂ Q {\displaystyle P=-{\frac {\partial F_{3}}{\partial Q}}\,\!} F = F 4 ( p , P , t ) = F 1 − q p + Q P {\displaystyle {\begin{aligned}F&=F_{4}(p,P,t)\\&=F_{1}-qp+QP\end{aligned}}} q = − ∂ F 4 ∂ p {\displaystyle q=-{\frac {\partial F_{4}}{\partial p}}\,\!} and Q = ∂ F 4 ∂ P {\displaystyle Q=~~{\frac {\partial F_{4}}{\partial P}}\,\!}

## Example

Sometimes a given Hamiltonian can be turned into one that looks like the [harmonic oscillator](/source/Harmonic_oscillator) Hamiltonian, which is

H = a P 2 + b Q 2 . {\displaystyle H=aP^{2}+bQ^{2}.}

For example, with the Hamiltonian

H = 1 2 q 2 + p 2 q 4 2 , {\displaystyle H={\frac {1}{2q^{2}}}+{\frac {p^{2}q^{4}}{2}},}

where p is the generalized momentum and q is the [generalized coordinate](/source/Generalized_coordinates), a good canonical transformation to choose would be

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

This turns the Hamiltonian into

H = Q 2 2 + P 2 2 , {\displaystyle 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 ) . {\displaystyle F=F_{3}(p,Q).}

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

P = − ∂ F 3 ∂ Q , {\displaystyle P=-{\frac {\partial F_{3}}{\partial Q}},}

and substitute the expression for P from equation (**[1](#math_1)**), expressed in terms of p and Q:

p Q 2 = − ∂ F 3 ∂ Q {\displaystyle {\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](#math_1)**):

F 3 ( p , Q ) = p Q {\displaystyle F_{3}(p,Q)={\frac {p}{Q}}}

To confirm that this is the correct generating function, verify that it matches (**[1](#math_1)**):

q = − ∂ F 3 ∂ p = − 1 Q {\displaystyle q=-{\frac {\partial F_{3}}{\partial p}}={\frac {-1}{Q}}}

## See also

- [Hamilton–Jacobi equation](/source/Hamilton%E2%80%93Jacobi_equation)

- [Poisson bracket](/source/Poisson_bracket)

## References

1. **[^](#cite_ref-1)** Goldstein, Herbert; Poole, C. P.; Safko, J. L. (2001). *Classical Mechanics* (3rd ed.). Addison-Wesley. p. 373. [ISBN](/source/ISBN_(identifier)) [978-0-201-65702-9](https://en.wikipedia.org/wiki/Special:BookSources/978-0-201-65702-9).

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