Generating function (physics)

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.

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}}\,\!}

Sometimes a given Hamiltonian can be turned into one that looks like the 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, 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), 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):

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

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

• Hamilton–Jacobi equation
• Poisson bracket