Classical mechanics, part 5

This post has been migrated from my old blog, the math-physics learning seminar.

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.