# 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:

**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.

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.

## The Hamiltonian Function

## Hamiltonian Systems on Riemannian Manifolds

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