{{short description|Vector bundle of cotangent spaces at every point in a manifold}} In [[mathematics]], especially [[differential geometry]], the '''cotangent bundle''' of a [[smooth manifold]] is the [[vector bundle]] of all the [[cotangent space]]s at every point in the manifold. It may be described also as the [[dual bundle]] to the [[tangent bundle]]. This may be generalized to [[Category (mathematics)|categories]] with more structure than smooth manifolds, such as [[complex manifold]]s, or (in the form of cotangent sheaf) [[Algebraic variety|algebraic varieties]] or [[Scheme (mathematics)|schemes]]. In the smooth case, any Riemannian metric or symplectic form gives an isomorphism between the cotangent bundle and the tangent bundle, but they are not in general isomorphic in other categories.
== Formal definition via [[diagonal morphism]] == There are several equivalent ways to define the cotangent bundle. [[Cotangent sheaf#Construction through a diagonal morphism|One way]] is through a [[diagonal mapping]] <math>\Delta</math> and [[germ (mathematics)|germs]].
Let <math>M</math> be a [[Differentiable manifold|smooth manifold]] and let <math>M\times M</math> be the [[Cartesian product]] of <math>M</math> with itself. The [[diagonal mapping]] <math>\Delta</math> sends a point <math>p</math> in <math>M</math> to the point <math>(p,p)</math> of <math>M\times M</math>. The image of <math>\Delta</math> is called the diagonal. Let <math>\mathcal{I}</math> be the [[sheaf (mathematics)|sheaf]] of [[germ (mathematics)|germs]] of smooth functions on <math>M\times M</math> which vanish on the diagonal. Then the [[sheaf (mathematics)#Operations|quotient sheaf]] <math>\mathcal{I}/\mathcal{I}^2</math> consists of equivalence classes of functions which vanish on the diagonal modulo higher order terms. The [[cotangent sheaf]] is defined as the [[inverse image functor|pullback]] of this sheaf to <math>M</math>:
<math display="block">\Gamma T^*M=\Delta^*\left(\mathcal{I}/\mathcal{I}^2\right).</math>
By [[Taylor's theorem]], this is a [[locally free sheaf]] of modules with respect to the sheaf of germs of smooth functions of <math>M</math>. Thus it defines a [[vector bundle]] on <math>M</math>: the '''cotangent bundle'''.
[[Smooth function|Smooth]] [[Section (fiber bundle)|sections]] of the cotangent bundle are called (differential) [[one-form]]s.
== Contravariance properties ==
A smooth morphism <math> \phi\colon M\to N</math> of manifolds induces a [[Pullback (differential geometry)|pullback sheaf]] <math>\phi^*T^*N</math> on ''M''. There is an [[Pullback (differential geometry)#Pullback of cotangent vectors and 1-forms|induced map]] of vector bundles <math>\phi^*(T^*N)\to T^*M</math>.
== Examples == The tangent bundle of the vector space <math>\mathbb{R}^n</math> is <math>T\,\mathbb{R}^n = \mathbb{R}^n\times \mathbb{R}^n</math>, and the cotangent bundle is <math>T^*\mathbb{R}^n = \mathbb{R}^n\times (\mathbb{R}^n)^*</math>, where <math>(\mathbb{R}^n)^*</math> denotes the [[dual space]] of covectors, linear functions <math>v^*:\mathbb{R}^n\to \mathbb{R}</math>.
Given a smooth manifold <math>M\subset \mathbb{R}^n</math> embedded as a [[hypersurface]] represented by the vanishing locus of a function <math>f\in C^\infty (\mathbb{R}^n),</math> with the condition that <math>\nabla f \neq 0,</math> the tangent bundle is
:<math>TM = \{(x,v) \in T\,\mathbb{R}^n \ :\ f(x) = 0,\ \, df_x(v) = 0\},</math>
where <math>df_x \in T^*_xM</math> is the [[directional derivative]] <math>df_x(v) = \nabla\! f(x)\cdot v</math>. By definition, the cotangent bundle in this case is
:<math>T^*M = \bigl\{(x,v^*)\in T^*\mathbb{R}^n \ :\ f(x)=0,\ v^* \in T^*_xM \bigr\},</math> where <math>T^*_xM=\{v \in T_x\mathbb{R}^n\ :\ df_x(v)=0\}^*.</math> Since every covector <math>v^* \in T^*_xM</math> corresponds to a unique vector <math>v \in T_xM</math> for which <math>v^*(u) = v \cdot u,</math> for an arbitrary <math>u \in T_xM,</math>
:<math>T^*M = \bigl\{(x,v^*)\in T^*\mathbb{R}^n\ :\ f(x) = 0,\ v \in T_x\mathbb{R}^n,\ df_x(v)=0 \bigr\}.</math>
== The cotangent bundle as phase space ==
Since the cotangent bundle <math>X=T^*M</math> is a [[vector bundle]], it can be regarded as a manifold in its own right. Because at each point the tangent directions of <math>M</math> can be paired with their dual covectors in the fiber, <math>X</math> possesses a canonical one-form <math>\theta</math> called the [[tautological one-form]], discussed below. The [[exterior derivative]] of <math>\theta</math> is a [[Symplectic manifold|symplectic 2-form]], out of which a non-degenerate [[volume form]] can be built for <math>X</math>. For example, as a result <math>X</math> is always an [[orientable]] manifold (the tangent bundle <math>TX</math> 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 manifold]]s, any real function on the cotangent bundle can be interpreted to be a [[symplectic vector space|Hamiltonian]]; thus the cotangent bundle can be understood to be a [[phase space]] on which [[Hamiltonian mechanics]] plays out.
=== The tautological one-form === {{main|Tautological one-form}}
The cotangent bundle carries a canonical one-form <math>\theta</math> also known as the [[symplectic potential]], ''Poincaré'' ''1''-form, or ''Liouville'' ''1''-form. This means that if we regard <math>T^*M</math> as a manifold in its own right, there is a canonical [[Section (fiber bundle)|section]] of the vector bundle <math>T^*(T^*M)</math> over <math>T^*M</math>.
This section can be constructed in several ways. The most elementary method uses local coordinates. Suppose that <math>x^i</math> are local coordinates on the base manifold <math>M</math>. In terms of these base coordinates, there are fibre coordinates <math>p_i</math>: a one-form at a particular point of <math>T^*M</math> has the form <math>p_i\text{d}x^i</math> ([[Einstein summation convention]] implied). So the manifold <math>T^*M</math> itself carries local coordinates <math>(x^i,p_i)</math> where the <math>x</math>'s are coordinates on the base and the <math>p</math>'s are coordinates in the fibre. The canonical one-form is given in these coordinates by <math display="block">\theta_{(x,p)}=\sum_{i=1}^n p_i \, dx^i.</math>
Intrinsically, the value of the canonical one-form in each fixed point of <math>T^*M</math> is given as a [[pullback (differential geometry)|pullback]]. Specifically, suppose that <math>\pi:T^*M\rightarrow M</math> is the [[Projection (mathematics)|projection]] of the bundle. Taking a point in <math>T^*_xM</math> is the same as choosing of a point <math>x</math> in <math>M</math> and a one-form <math>\omega</math> at <math>x</math>, and the tautological one-form <math>\theta</math> assigns to the point <math>(x,\omega)</math> the value <math display="block">\theta_{(x,\omega)}=\pi^*\omega.</math>
That is, for a vector <math>v</math> in the tangent bundle of the cotangent bundle, the application of the tautological one-form <math>\theta</math> to <math>v</math> at <math>(x,\omega)</math> is computed by projecting <math>v</math> into the tangent bundle at <math>x</math> using <math>\text{d}\pi:T(T^*M)\rightarrow TM</math> and applying <math>\omega</math> to this projection. Note that the tautological one-form is not a pullback of a one-form on the base <math>M</math>.
=== Symplectic form ===
The cotangent bundle has a canonical [[Symplectic manifold|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 <math>\mathbb{R}^n \times \mathbb{R}^n</math>. But there the one form defined is the sum of <math>y_i\,dx_i</math>, and the differential is the canonical symplectic form, the sum of <math>dy_i \land dx_i</math>.
=== Phase space ===
If the manifold <math>M</math> represents the set of possible positions in a [[dynamical system]], then the cotangent bundle <math>T^{*}\!M</math> 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.
==See also== * [[Legendre transformation]]
== References ==
* {{cite book |authorlink=Ralph Abraham (mathematician) |first=Ralph |last=Abraham |authorlink2=Jerrold E. Marsden |first2=Jerrold E. |last2=Marsden |title=Foundations of Mechanics |year=1978 |publisher=Benjamin-Cummings |location=London |isbn=0-8053-0102-X }} * {{cite book |first=Jürgen |last=Jost |author-link=Jürgen Jost |title=Riemannian Geometry and Geometric Analysis |year=2002 |publisher=Springer-Verlag |location=Berlin |isbn=3-540-63654-4 }} * {{cite book |first=Stephanie Frank |last=Singer |title=Symmetry in Mechanics: A Gentle Modern Introduction|title-link= Symmetry in Mechanics |year=2001 |publisher=Birkhäuser |location=Boston }}
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[[Category:Vector bundles]] [[Category:Differential topology]] [[Category:Tensors]]