For those interested, I thought I would give a very brief overview of the Hamiltonian approach to classical mechanics.

The idea is to treat the classical path of a particle as a curve in phase space--i.e., as a function (x(t), p(t)), where x = position, p = momentum. In the 1-d situation, phase space is the set of all points (x,p), which is isomorphic to the set of all 1-forms on R:

There is a symplectic 2-form on phase space naturally associated to this structure: Namely

In general, we may consider phase space to be any symplectic manifold. If motion is confined to move on some manifold M, then we should take phase space to be the symplectic manifold T*M, i.e., the set of all 1-forms on M. T*M has a natural symplectic 2-form obtained by extending the definition given above.

Let g be any function on phase space. Using the fact that the symplectic form is non-degenerate, we can find a vectorfield

such that

Let H be a function on phase space such that the is invariant under the flow of . We think of the vectorfield generated by H as time translation. In general, for any function g on phase space, we define to be the time derivative of g. Let

.

Then we have the dynamics:

Note H' = {H,H} = 0. So H, the energy, is conserved.

The Hamiltonian formalism is particularly useful for passing into quantum mechanics. To obtain the quantum version of the above classical system, you declare that all functions g on phase space are self-adjoint operators on some Hilbert space. Pretty much everything you need to know can be figured out from the commutation relations of the operators. In the quantum theory, we let

(In units where planck's constant is 1). Everything in classical quantum theory follows from this. Even the Schrodinger equation is just a consequence of the fact that

as operators.

Unfortunately, I have barely touched on this subject here. But I wanted to give a brief snapshot of the subject.