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 $T^*M$.

As in the case $M = \mathbb{R}^n$, we will still call the phase space coordinates by $(\mathbf{x}, \mathbf{p})$ (although commonly $\mathbf{q}$ is used in place of $\mathbf{x}$). The first coordinate $\mathbf{x}$ takes values in the manifold $M$ and represents the position of the particle. The most difficult part of extending Hamiltonian systems to manifolds is understanding what is the natural home of the second variable $\mathbf{p}$ is. In the language of mathematics the above says that $\mathbf{p}$ is a linear functional on the space of velocity vectors for the manifold $M$. In physics one thinks of such functionals as a type of momentum, and in the Hamiltonian systems literature the variable $\mathbf{p}$ is frequently referred to as momentum (or more precisely, a conjugate momentum). Mathematically speaking the conjugate momentum is a linear functional on velocity.

Let’s briefly step back and think about “velocity” for a particle moving on a manifold. As the particle moves it has a velocity vector at each time. For a surface embedded into some $\mathbb{R}^N$, the velocity vector can be regarded as a vector that is tangent to the surface and rooted at the current position of the particle. In the framework of manifolds the natural home of “velocity vectors” is the tangent space and the tangent bundle. At a point $\mathbf{x} \in M$ the tangent space is called $T_{\mathbf{x}} M$, and is a linear subspace attached to the point $\mathbf{x}$. The elements of $T_{\mathbf{x}} M$ represent the velocity vectors at the point $\mathbf{x}$. There are several equivalent ways to think of these vectors: probably my favorite is as the derivatives of curves on $M$ as they pass through $\mathbf{x}$.

Since $M$ is assumed to be $n$-dimensional each tangent space $T_{\mathbf{x}} M$ will also be $n$-dimensional, and isomorphic to a copy of $\mathbb{R}^n$. Moreover every choice of a local coordinate system around a point $\mathbf{x} \in M$ induces a basis for the linear space $T_{\mathbf{x}} M$. You can think of the basis in this way:

  • a local coordinate system around $\mathbf{x}$ just means a collection of $n$ variables (i.e. a subset of $\mathbb{R}^n$) such that each point in the collection gets uniquely mapped to a point in $M$ (in other words, the local coordinate system is just a diffeomorphism $\phi : U \to \mathbb{R^n}$ where $U$ is an open subset of $M$ containing $\mathbf{x}$ - this is typically called a coordinate chart),
  • this induces $n$ “special” curves passing through $\mathbf{x}$, each curve being a level line of $n-1$ of the coordinates with the remaining coordinate changing in some pre-determined way (say linearly)
  • the basis for $T_{\mathbf{x}} M$ is simply the derivatives of these $n$ special curves.

As sets the tangent bundle $TM$ and the cotangent bundle $T^*M$ are simple to describe $$ TM = \{ (\mathbf{x}, \mathbf{y}) : \mathbf{x} \in M, \mathbf{y} \in T_{\mathbf{x}} M \} $$ and similarly $$ T^*M = \{ (\mathbf{x}, \mathbf{p}) : \mathbf{x} \in M, \mathbf{p} \in T_{\mathbf{x}}^* M \} $$ Of course one main purpose of the theory of manifolds is to isolate the constructions and results that *do not* depend on any particular coordinate system, but I mention the above since it is a nice example of how the data on $M$ (in this case the local coordinate system) naturally extends to extra data on $T_{\mathbf{x}} M$ (in this case the basis of the space).

The cotangent space then enters into the picture quite naturally.

Both the tangent bundle $TM$ and the cotangent bundle $T^* M$ are manifolds in their own right. If $M$ is $n$-dimensional then both $TM$ and $T^* M$ are $2n$-dimensional manifolds.

This statement is useful for bringing manifold theory into the discussion. A set $M$ becomes a manifold by equipping it with some extra data - essentially an atlas of coordinate charts that allow us to extend a notion of smoothness to real-valued functions on $M$. The statement above says that the same extra data can be used to extend $TM$ and $T^* M$ from sets to manifolds, modulo some lengthy constructions which we don’t discuss here.

Note that formulating the phase space as $T^* M$ matches the phase space for $M = \mathbb{R}^n$. When $M = \mathbb{R}^n$ then $M$ and the tangent space and the co-tangent spaces are all the same ($\mathbb{R}^n$ itself), but there is no real way to see how the objects are different.

The Hamiltonian Function

Hamiltonian Systems on Riemannian Manifolds

A manifold becomes Riemannian when it is equipped with some extra data on the tangent bundle.