Thread: Hamiltonian mechanics

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.

Sorry the TeX isn't formatting. I think there's a problem with the java script or something.

So, basically your saying that "straight line" paths dont exist? This is quite true in a certain way because Huygens principal shows that all waves are actually spherical and just create a wave front.

No, I never said there were no straight line paths. Hamiltonian mechanics predicts the same results as Newtonian mechanics, it's just a different methodology. So, in particular, if the background potential energy is zero, the particle will move in a straight line.

I think what it is is that hamiltonian mechanics is more useful when you aren't dealing with regular, easy to compute, curves. Straight lines and circular arcs would be relatively simple using netonian mechanics, and the other methods wouldn't really be necessary. At least, thats what I'm getting out of these two threads.

Most importantly, the Lagrangian and Hamiltonian approach suggests the proper way to pass from a classical system to its quantum counterpart. Simply promote observables to operators, where the commutator of two operators is i*Poisson bracket.



The Hamiltonian and Lagrangian approach also shed light on where conservation laws in nature really come from. There's a theorem called "Noether's Theorem" that says that continuous symmetries of the Lagrangian give rise to conserved quantities. In particular:

Translational symmetry----> Conservation of momentum

Rotational symmetry-----> Conservation of angular momentum

Time translational symmetry----> Conservation of Energy

This correspondence between symmetries and conserved quantities becomes especially important in quantum field theory. There we find, to give one example:

Symmetry in angular phase of quantum field---> Conservation of charge

this must be one of your favorite topics in physics.

This stuff is a little too fundamental to consider it a "favorite topic."

I don't know what my favorite topic would be. I'm still learning the subject. One subject that I've been trying to learn a little about is solitons, e.g., vortices and monopoles.

In quantum field theory, the classical equations of motion allow for some strange higher energy states that cannot deform to lower energy states due to topological constraints. So, for example, you can get things like vortices, which are 1-dim "strings" of concentrated energy that can stretch out to infinity. (No relation to the strings of string theory). Studying the space of all solitons turns out to be very useful in mathematics. Whether giant strings of energy, etc., actually exist of course is highly speculative.