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So in classical mechanics, we just went over the Hamiltonian formulation, Louisville, pretty basic stuff. But one thing that kept irking me, especially when it came to phase space and the relation that
df/dt = -{f,H} + part f/part t for any smooth f(q,p),
it seemed like we kept implicitly assuming that part H/part t = 0. I say this mostly 'cos the formulation of phase space as 2n-dimensional and the use of the 2n variables q and p in the relation
{f,g} = sum_(i) [(part f/part q_i)(part g/part p_i) - (part f/part p_i)(part g/partq_i)
doesn't mention time explicitly, as we'd assume the Hamiltonian to.

Now, it's not that I doubt that these relations are correct, but is that because of a property of the Poisson brackets? Do the Poisson brackets have to make use of every explicit variable?

So the fact that time is never mentioned explicitly is a property of Hamilton's equations, not the Poisson brackets. The transformation into p and q takes care of that.

Actually, our professor just gave us notes on this one (we're using Hand and Finch as the main text, which doesn't really cover Hamiltonians very well). He doesn't want us to get "bogged down in formal mathematics when approximations will get you the same answer just as quickly." So we had been taught that the Hamiltonian is a Legendre transformation of the Euler-Lagrange equations. He mentioned nothing of configuration space, tangent or cotangent space, and (like you said) just described everything as a coordinate transformation. I was a bit peeved that he also never mentioned momentum space, then proceeded to give us a homework problem on it. Not that it was too hard to figure out, but he could've mentioned it.

So then...configuration space? R? Cotangent space (I understand tangent space, but not cotangent space)?

Curses...I was so close to understanding until you said symplectic manifold. But I think I get the general idea:

The positions form the configuration space C = {q_i}; the collection of tangent spaces of C is TC = {T_(q_i) C}. The Hamiltonian operates on the dual space of this, T*C = {f: TC -> R}; H:T*C -> R. Where does phase space come in?

Hmm... I finally have the time to give it the treatment it deserves, since you were previously interested in differential geometry. This is quite a pain without TeX.

Recall that an n-dimensional manifold is a space that is locally homeomorphic to R^n. An atlas of coordinate charts {(x_α,U_α)} is a collection of open sets on U on the manifold and coordinate functions x from that U to R^n. Some overlaps between the open sets in the coordinate charts must occur for the manifold to be connected; these overlaps "live" in two different coordinate charts. The "transition functions" between those coordinate charts might be differentiable or even infinitely-differentiable (smooth), which is what is meant by a "smooth manifold". For now, let's treat this as a given (more general definitions of tangent and cotangent bundles are possible, without dependence on smoothness, but this particular version is much more easily translatable to classical mechanics).

Given a smooth n-dimensional manifold C, a tangent vector v at q in C is a linear operator on smooth real functions [1] that is also a linear derivation at q, that is, v satisfies v(fg) = f(q)v(g) + g(q)v(f). For example, given a coordinate chart (x,U), the operator [∂/∂x^k]|_q is a derivation at q. It is clear that the space of all linear derivations at q forms a vector space, which is called the tangent space at q and is denoted by T_qC.

It can also be shown that this vector space is spanned by {[∂/∂x^k]|_q: k=1,...,n}; hence, it is isomorphic to R^n and its elements are representable with coordinates in the natural way. The tangent bundle TC is a manifold with points {(q,q'):q in C, q' in T_qC} and structure defined as follows: for each coordinate chart (x,U) on C, define a coordinate chart (X,TU) on TC by TU = {(q,q'):q in U, q' in T_pC} and X((q,q')) = (x^1,...,x^n;q'^1,...,q'^n), where (q'^1,...,q'^n) is the coordinate representation of v in the coordinate system found by taking {[∂/∂x^k]|_q} as the basis. These are the standard coordinates for the Lagrangian formulation of classical mechanics.

The canonical projection π:TC→C is defined by (q,v)↦q, with the individual tangent spaces π^{-1}(q) = T_qC being the fibres of the bundle. Additionally, for any open U⊂C, a continuous f:U→TC satisfying (π o f)(q) = q is a local section.

Given a smooth f, we may define df_q:T_qC→R by df_q(X_q) = X_q(f); the operator df_q is a linear functional on T_qC, i.e., an element of the dual space T_q^*C, which is called the cotangent space at q. Thus, for any coordinate chart (x,U), a local section df can be defined in this manner, and is called the differential of f. In particular, the differentials [dx^k]|_q define a basis for the cotagent space T_q^*C, so that the complete cotangent bundle T^*C can be constructed in the same manner as TC. If we call the corresponding coordinates p_k, then these are the canonical coordinates for the Hamiltonian formulation of classical mechanics.

An example: X_f|(p,q) = {f, } for any smooth f on the phase space T^*C is a tangent vector on the phase space itself, since this Poisson bracket is composed of partials on the phase space (seen previously, these form a basis on the tangent space). Thus, X_f is a vector field on the phase space. In the particular case of the Hamiltonian itself, X = X_H is the Hamiltonian vector field that dictates the evolution of every observable (actually, there are two conventions, either {H, } or -{H, } = { ,H}). In terms of differential geometry, Xf = {H,f} is the Lie derivative of the observable f in the direction of the Hamiltonian vector field X = X_H.


[1] That is, a function f:C→R is smooth iff for every p in C and every coordinate chart (x,U) for which p in U, the coordinate function x:U→R^n has (f o x^{-1}):R^n→R infinitely often differentiable at x(p) in the usual sense.