Classical Mechanics 5: Symplectic structures

As we saw in the previous post, the equations of motion for a mechanical system can be cast into a 1st order form called Hamilton's equations, which are naturally interpreted as describing a path in the phase space $T^\ast M$ associated to the configuration space $M$. Let us investigate the geometry of $T^\ast M$ see why Hamilton's equations are so nice.

Definition The canonical (or sometimes tautological) 1-form on the cotangent bundle $T^\ast M$ is the 1-form $\theta$ defined by
$$\theta_{(q,p)}(X) = p(\pi_\ast X),$$
where $\pi_\ast$ is the pushforward induced by the natural projection $\pi: T^\ast M \to TM$. In other words, the form is defined by
$$\theta_{(q,p)} = \pi^\ast p.$$

Definition The canonical symplectic form on the cotangent bundle $T^\ast M$ is the 2-form $\omega$ defined by
$$\omega = -d\theta.$$

Let $\omega_\flat: T M \to T^\ast M$ be the map given by $X \mapsto \iota(X)\omega$.

Proposition The canonical symplectic form satisfies the following two conditions:
1. It is closed, i.e. $d\omega = 0$.
2. It is nondegenerate, i.e. the map $\omega_\flat$ is invertible with inverse $\omega^\sharp: T^\ast M \to TM$.

Proof The first property follows from $d^2 = 0$. To prove the second, suppose we have local coordinates $q^i$ on $M$ with cotangent coordinates $p^i$. Then it is easily seen that
$$\theta = p^i dq^i,$$
so that
$$\omega = dq^i \wedge dp^i,$$
from which nondegeneracy is obvious.

Definition Any 2-form on a manifold $N$ (not necessarily a cotangent bundle) which satisfies the above two properties will be called symplectic. A pair $(N, \omega)$ will be called symplectic if $\omega$ is a symplectic 2-form on $N$.

Definition Given a function $H$ on a symplectic manifold $(N, \omega)$, the Hamiltonian vector field associated to $H$ is the vector field $X_H$ uniquely defined by
$$dH = \omega_\flat X_H.$$

Proposition For $N = T^\ast M$ a cotangent bundle with the canonical symplectic form, Hamilton's equations with respect to a Hamiltonian function $H$ describe the flow of the vector field $X_H$.

Proof Again pick local coordinates $q$ and $p$. Then the inverse map $\omega^\sharp$ is given by
$$dq \mapsto -\frac{\partial}{\partial p}$$
$$dp \mapsto \frac{\partial}{\partial q}$$
Since
$$dH = \frac{\partial H}{\partial q} dq + \frac{\partial H}{\partial p} dp,$$
we see that
$$X_H = \frac{\partial H}{\partial p} \frac{\partial}{\partial q} - \frac{\partial H}{\partial q} \frac{\partial}{\partial p}$$
But then the equation describing the flow of $X_H$ is (in components)
$$\dot{q} = \frac{\partial H}{\partial p}$$
$$\dot{p} = -\frac{\partial H}{\partial q}$$
which are exactly Hamilton's equations.