Two things today motivated this question.
First, the professor said that in a lecture Thurston mentioned
Any manifold can be seen as the configuration space of some physical system.
Clearly we need to be careful here, so the first question is
1) What is a precise formulation and an argument to see why the previous statement is true.
Second, the professor went on to say that because of the Poisson Bracket, we see the phase space of a physical system as the Cotangent Bundle of a manifold. I understand that we associate a symplectic form to the cotangent bundle, and that we want to think of Phase Space with a symplectic structure, but my second question is
2) Could you provide an example of a physical system, give the associated "configuration manifold" show the cotangent space, and explain why this is the phase space of the system.
I pushed the lecturer quite a bit to get this level of detail, so pushing much further would of probably been considered rude. I should also mention he is speaking about these things because we want to quantize the geometry associated to this manifold. So he looks at the cotangent space which apparently has a symplectic structure, and sends the symplectic form to the Lie Bracket. If any of the things I have said are incorrect, please comment with corrections. I am just learning this material and trying to understand how it fits with my current understanding.
One last bonus question(tee hee),
If your manifold is a Lie Group, we get a Lie algebra structure on the Tangent Space. Is there a relationship between this Lie Algebra structure and the one you would get by considering the cotangent space and then quantizing in the fashion of above?
Thanks in advance!
Best Answer
Let's start by answering the first question.
Let $M$ be any manifold. Consider a physical system consisting of a point-particle moving on $M$. What are the configurations of this physical system? The points of $M$. Hence $M$ is the configuration space.
Typically one takes $M$ to be riemannian and we may add a potential function on $M$ in order to define the dynamics. (More complicated dynamics are certainly possible -- this is just the simplest example.)
As an example, let's consider a point particle of mass $m$ moving in $\mathbb{R}^3$ under the influence of a central potential $$V= k/r,$$ where $r$ is the distance from the origin. The configuration space is $M = \mathbb{R}^3\setminus\lbrace 0\rbrace$.
Classical trajectories are curves $x(t)$ in $M$ which satisfy Newton's equation $$m \frac{d^2 x}{dt^2} = \frac{k}{|x|^2}.$$ To write this equation as a first order equation we introduce the velocity $v(t) = \frac{dx}{dt}$. Geometrically $v$ is a vector field (a section of the tangent bundle $TM$) and hence the classical trajectory $(x(t),v(t))$ defines a curve in $TM$ satisfying a first order ODE: $$\frac{d}{dt}(x(t),v(t)) = (v(t), \frac{k}{m|x(t)|^2})$$ This equation can be derived from a variational problem associated to a lagrangian function $L: TM \to \mathbb{R}$ given by $$L(x,v) = \frac12 m v^2 - \frac{k}{|x|}.$$ The fibre derivative of the lagrangian function defines a bundle morphism $TM \to T^*M$: $$(x,v) \mapsto (x,p)$$ where $$p(x,v) = \frac{\partial L}{\partial v}.$$
In this example, $p = mv$. The Legendre transform of the lagrangian function $L$ gives a hamiltonian function $H$ on $T^*M$, which in this example is the total energy of the system: $$H(x,p) = \frac{1}{2m}p^2 + \frac{k}{|x|}.$$
The equations of motion can be recovered as the flow along the hamiltonian vector field associated to $H$ via the standard Poisson brackets in $T^*M$:
$$ \frac{dx}{dt} = \lbrace x,H \rbrace \qquad\mathrm{and}\qquad \frac{dp}{dt} = \lbrace p,H \rbrace.$$
Being integral curves of a vector field, there is a unique classical trajectory through any given point in $T^*M$, hence $T^*M$ is a phase space for the system; that is, a space of states of the physical system. Of course $TM$ is also a space of states, but historically one calls $T^*M$ the phase space of the system with configuration space $M$. (I don't know the history well enough to know why. There are brackets in $TM$ as well and one could equally well work there.)
Not every space of states is a cotangent bundle, of course. One can obtain examples by hamiltonian reduction from cotangent bundles by symmetries which are induced from diffeomorphisms of the configuration space, for instance. Or you could consider systems whose physical trajectories satisfy an ODE of order higher than 2, in which case the cotangent bundle is not the space of states, since you need to know more than just the position and the velocity at a point in order to determine the physical trajectory.
It's late here, so I'll forego answering the bonus question for now.