Cotangent bundle

Contravariance properties

A smooth morphism ${\displaystyle \varphi :M\to N}$ of manifolds induces a pullback sheaf ${\displaystyle {\varphi }^{\ast }{T}^{\ast }N}$ on M. There is an induced map of vector bundles ${\displaystyle {\varphi }^{\ast }(T^{\ast }N)\to T^{\ast }M}$.

Examples

The tangent bundle of the vector space ${\displaystyle {\mathbb{R}}^n}$ is ${\displaystyle T{\mathbb{R}}^n={\mathbb{R}}^n\times {\mathbb{R}}^n}$, and the cotangent bundle is ${\displaystyle T^{\ast }{\mathbb{R}}^n={\mathbb{R}}^n\times ({\mathbb{R}}^n{)}^{\ast }}$, where ${\displaystyle ({\mathbb{R}}^n{)}^{\ast }}$ denotes the dual space of covectors, linear functions ${\displaystyle v^{\ast }:{\mathbb{R}}^n\to \mathbb{R}}$.

Given a smooth manifold ${\displaystyle M\subset {\mathbb{R}}^n}$ embedded as a hypersurface represented by the vanishing locus of a function ${\displaystyle f\in C^{\mathrm{\infty }}({\mathbb{R}}^n),}$ with the condition that ${\displaystyle \mathrm{\nabla }f\ne 0,}$ the tangent bundle is

${\displaystyle TM=\{(x,v)\in T{\mathbb{R}}^n:f(x)=0,d{f}_x(v)=0\},}$

where ${\displaystyle d{f}_x\in T_x^{\ast }M}$ is the directional derivative ${\displaystyle d{f}_x(v)=\mathrm{\nabla }f(x)\cdot v}$. By definition, the cotangent bundle in this case is

${\displaystyle T^{\ast }M=\{(x,v^{\ast })\in T^{\ast }{\mathbb{R}}^n:f(x)=0,v^{\ast }\in T_x^{\ast }M\},}$

where ${\displaystyle T_x^{\ast }M=\{v\in T_x{\mathbb{R}}^n:d{f}_x(v)=0{\}}^{\ast }.}$ Since every covector ${\displaystyle v^{\ast }\in T_x^{\ast }M}$ corresponds to a unique vector ${\displaystyle v\in T_{x}M}$ for which ${\displaystyle v^{\ast }(u)=v\cdot u,}$ for an arbitrary ${\displaystyle u\in T_{x}M,}$

${\displaystyle T^{\ast }M=\{(x,v^{\ast })\in T^{\ast }{\mathbb{R}}^n:f(x)=0,v\in T_x{\mathbb{R}}^n,d{f}_x(v)=0\}.}$

The cotangent bundle as phase space

Since the cotangent bundle ${\displaystyle X=T^{\ast }M}$ is a vector bundle, it can be regarded as a manifold in its own right. Because at each point the tangent directions of ${\displaystyle M}$ can be paired with their dual covectors in the fiber, ${\displaystyle X}$ possesses a canonical one-form ${\displaystyle \theta }$ called the tautological one-form, discussed below. The exterior derivative of ${\displaystyle \theta }$ is a symplectic 2-form, out of which a non-degenerate volume form can be built for ${\displaystyle X}$. For example, as a result ${\displaystyle X}$ is always an orientable manifold (the tangent bundle ${\displaystyle TX}$ is an orientable vector bundle). A special set of coordinates can be defined on the cotangent bundle; these are called the canonical coordinates. Because cotangent bundles can be thought of as symplectic manifolds, any real function on the cotangent bundle can be interpreted to be a Hamiltonian; thus the cotangent bundle can be understood to be a phase space on which Hamiltonian mechanics plays out.

The tautological one-form

The cotangent bundle carries a canonical one-form ${\displaystyle \theta }$ also known as the symplectic potential, Poincaré 1-form, or Liouville 1-form. This means that if we regard ${\displaystyle T^{\ast }M}$ as a manifold in its own right, there is a canonical section of the vector bundle ${\displaystyle T^{\ast }(T^{\ast }M)}$ over ${\displaystyle T^{\ast }M}$.

This section can be constructed in several ways. The most elementary method uses local coordinates. Suppose that ${\displaystyle x^i}$ are local coordinates on the base manifold ${\displaystyle M}$. In terms of these base coordinates, there are fibre coordinates ${\displaystyle p_i}$: a one-form at a particular point of ${\displaystyle T^{\ast }M}$ has the form ${\displaystyle p_i\text{d}{x}^i}$ (Einstein summation convention implied). So the manifold ${\displaystyle T^{\ast }M}$ itself carries local coordinates ${\displaystyle (x^i,p_i)}$ where the ${\displaystyle x}$'s are coordinates on the base and the ${\displaystyle p}$'s are coordinates in the fibre. The canonical one-form is given in these coordinates by $${\displaystyle {\theta }_{(x,p)}=\sum _{i=1}^n{p}_{i}d{x}^i.}$$

Intrinsically, the value of the canonical one-form in each fixed point of ${\displaystyle T^{\ast }M}$ is given as a pullback. Specifically, suppose that ${\displaystyle \pi :T^{\ast }M\to M}$ is the projection of the bundle. Taking a point in ${\displaystyle T_x^{\ast }M}$ is the same as choosing of a point ${\displaystyle x}$ in ${\displaystyle M}$ and a one-form ${\displaystyle \omega }$ at ${\displaystyle x}$, and the tautological one-form ${\displaystyle \theta }$ assigns to the point ${\displaystyle (x,\omega )}$ the value $${\displaystyle {\theta }_{(x,\omega )}={\pi }^{\ast }\omega .}$$

That is, for a vector ${\displaystyle v}$ in the tangent bundle of the cotangent bundle, the application of the tautological one-form ${\displaystyle \theta }$ to ${\displaystyle v}$ at ${\displaystyle (x,\omega )}$ is computed by projecting ${\displaystyle v}$ into the tangent bundle at ${\displaystyle x}$ using ${\displaystyle \text{d}\pi :T(T^{\ast }M)\to TM}$ and applying ${\displaystyle \omega }$ to this projection. Note that the tautological one-form is not a pullback of a one-form on the base ${\displaystyle M}$.

Symplectic form

The cotangent bundle has a canonical symplectic 2-form on it, as an exterior derivative of the tautological one-form, the symplectic potential. Proving that this form is, indeed, symplectic can be done by noting that being symplectic is a local property: since the cotangent bundle is locally trivial, this definition need only be checked on ${\displaystyle {\mathbb{R}}^n\times {\mathbb{R}}^n}$. But there the one form defined is the sum of ${\displaystyle y_{i}d{x}_i}$, and the differential is the canonical symplectic form, the sum of ${\displaystyle d{y}_i\wedge d{x}_i}$.

Phase space

If the manifold ${\displaystyle M}$ represents the set of possible positions in a dynamical system, then the cotangent bundle ${\displaystyle T^{\ast }M}$ can be thought of as the set of possible positions and momenta. For example, this is a way to describe the phase space of a pendulum. The state of the pendulum is determined by its position (an angle) and its momentum (or equivalently, its velocity, since its mass is constant). The entire state space looks like a cylinder, which is the cotangent bundle of the circle. The above symplectic construction, along with an appropriate energy function, gives a complete determination of the physics of system. See Hamiltonian mechanics and the article on geodesic flow for an explicit construction of the Hamiltonian equations of motion.