Today, over lunch, togther with Christian Römmelsberger, we tried to understand Hamilton-Jacobi theory from a more geometric point of view.
The way this is usually presented (in the very end of a course on classical mechanics) is in terms of generating functions for canonical transformations such that in the new coordinates the Hamiltonian vanishes. Here I will rewrite this in the laguage of symplectic geometry.
As always, let us start with a 2N dimensional symplectic space M with symplectic form . In addition, pick N functions such that the submanifolds are Lagrangian, that is is a Lagrangian subspace of (meaning that the symplectic form of any two tangent vectors of vanishes). If this holds, the can be regarded as position coordinates.
Starting from these Lagrangian submanifolds, we can locally find a family of 1-forms in the normal bundle such that . You should think that for appropriate momentum coordinates on the Lagrangian leaves of constant . But here, these are just coefficient funtions to make a potential for .
Now we repeat this for another set of position coordinates which we assume to be "sufficiently independent" of the meaning that . This implies that locally are coordinates on M. With the comes another 1-form and since are locally related by a "gauge transformation". We have for a function F.
Let's look at a little bit closer. A general normal 1-form would look like . But since we started from Lagrangian leaves, there is no in and thus . But expressing this in coordinates yields
Comparing coefficients we find and . You will recognize the expressions for momenta in terms of a "generating function".
What we have done was to take two Lagrangian foliations given in terms of and and compute a function from them. The trick is now to turn this procedure around: Given only the and a function of these and some N other variables , one can compute the as functions on M: Take a point and define by inverting . For this remember that was defined implicitly above: It is the coefficient of in .
Up to here, we have only played symplectic games independent of any dynamics. Now specify this in addition in terms of a Hamilton function h. Then the Hamilton-Jacobi equations are nothing but the requirement to find a generating function such that the are constants of motion.
Even better, by making everything (that is h and Q and F) explicitly time dependent, by the requirement that the action 1-form is invariant: giving we get a transforming Hamilonian and we can require this to vanish:
If we think of the Hamiltonian h given in terms of the coordinates this is now a PDE for F which has to hold for all . That is, writing F as a function of and it has to hold for all (fixed) as a PDE in the and t.