10.4: Generating Functions in Different Variables

This $\begin{equation} F(q, Q, t) \end{equation}$ is only one example of a generating function—in discussing Liouville’s theorem later, we’ll find it convenient to have a generating function expressed in the q 's and P 's. We get that generating function, often labeled $\begin{equation} \Phi(q, P, t), \text { from } F(q, Q, t) \end{equation}$ by a Legendre transformation:

$$\begin{equation} d \Phi(q, P, t)=d\left(F+\sum P_{i} Q_{i}\right)=\sum p_{i} d q_{i}+\sum Q_{i} d P_{i}+\left(H^{\prime}-H\right) d t \end{equation}$$

Then, for this new generating function

$$\begin{equation} p_{i}=\dfrac{\partial \Phi(q, P, t)}{\partial q_{i}}, \quad Q_{i}=\dfrac{\partial \Phi(q, P, t)}{\partial P_{i}}, \quad H^{\prime}=H+\dfrac{\partial \Phi(q, P, t)}{\partial t} \end{equation}$$

Evidently, we can similarly use the Legendre transform to find generating functions depending on the other possible mixes of old and new variables: p,Q, and p,P.

What’s the Point of These Canonical Transformations?

It will become evident with a few examples: it is often possible to transform to a set of variables where the equations of motion are a lot simpler, and, for some variables, trivial. The canonical approach also gives a neat proof of Liouville’s theorem, which we’ll look at shortly.