Classical mechanics, part 5

This post has been migrated from my old blog, the math-physics learning seminar. The original post can be found here.

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.