Hamiltonian Systems on Manifolds

This article briefly explains how a Hamiltonian system is described on a manifold. Throughout we let $M$ be an $n$-dimensional manifold, for our purposes it is sufficient to think of it as an $n$-dimensional surface embedded in some higher dimensional space. Think of the $2$-sphere or the $2$-torus living in $3$-dimensional space. Everything we will describe also works for abstract manifolds, as long as it’s equipped with a differentiable structure.

Phase Space

The Hamiltonian formulation describes the motion of a particle along the manifold, but the manifold itself is not the phase space of the system. Recall that when the manifold is $\mathbb{R}^n$ the evolution equation actually takes place on $\mathbb{R}^{2n}$, i.e. the phase space is $\mathbb{R}^{2n}$. This is because the Hamiltonian function specifies a coupled system of two first-order equations. The main purpose of this article is to explain what the phase space is for a Hamiltonian system on a manifold. The short answer:

For a Hamiltonian system on an $n$-dimensional manifold $M$, the phase space of the system is the cotangent bundle $T^*M$.