Editing Cotangent bundle

{{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]] &Delta; and  [[germ (mathematics)|germs]].

Let ''M'' be a [[Differentiable manifold|smooth manifold]] and let ''M''&times;''M'' be the [[Cartesian product]] of ''M'' with itself.  The [[diagonal mapping]] &Delta; sends a point ''p'' in ''M'' to the point (''p'',''p'') of ''M''&times;''M''.  The image of &Delta; is called the diagonal.  Let <math>\mathcal{I}</math> be the [[sheaf (mathematics)|sheaf]] of [[germ (mathematics)|germs]] of smooth functions on ''M''&times;''M'' 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 ''M'':

:<math>\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 ''M''.  Thus it defines a [[vector bundle]] on ''M'':  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 ''X'' = ''T''*''M'' is a [[vector bundle]], it can be regarded as a manifold in its own right.  Because at each point the tangent directions of ''M'' can be paired with their dual covectors in the fiber, ''X'' possesses a canonical one-form &theta; called the [[tautological one-form]], discussed below. The [[exterior derivative]] of &theta; is a [[symplectic form|symplectic 2-form]], out of which a non-degenerate [[volume form]] can be built for ''X''.  For example, as a result ''X'' is always an [[orientable]] manifold (the tangent bundle ''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 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 &theta; also known as the [[symplectic potential]], ''Poincaré'' ''1''-form, or ''Liouville'' ''1''-form. This means that if we regard ''T''*''M'' as a manifold in its own right, there is a canonical [[Section (fiber bundle)|section]] of the vector bundle ''T''*(''T''*''M'') over ''T''*''M''.

This section can be constructed in several ways.  The most elementary method uses local coordinates.  Suppose that ''x''<sup>''i''</sup> are local coordinates on the base manifold ''M''.  In terms of these base coordinates, there are fibre coordinates ''p''<sub>''i''</sub> :  a one-form at a particular point of ''T''*''M'' has the form ''p''<sub>''i''</sub>&nbsp;''dx''<sup>''i''</sup> ([[Einstein summation convention]] implied).  So the manifold ''T''*''M'' itself carries local coordinates (''x''<sup>''i''</sup>, ''p''<sub>''i''</sub>) where the ''x''<nowiki/>'s are coordinates on the base and the ''p's'' are coordinates in the fibre.  The canonical one-form is given in these coordinates by

:<math>\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 ''T*M'' is given as a [[pullback (differential geometry)|pullback]].  Specifically, suppose that {{nowrap|&pi; : ''T*M'' &rarr; ''M''}} is the [[Projection (mathematics)|projection]] of the bundle. Taking a point in ''T''<sub>''x''</sub>*''M'' is the same as choosing of a point ''x'' in ''M'' and a one-form &omega; at ''x'', and the tautological one-form &theta; assigns to the point (''x'', &omega;) the value

:<math>\theta_{(x,\omega)}=\pi^*\omega.</math>

That is, for a vector ''v'' in the tangent bundle of the cotangent bundle, the application of the tautological one-form &theta; to ''v'' at (''x'', &omega;) is computed by projecting ''v'' into the tangent bundle at ''x'' using {{nowrap|''d''&pi; : ''T''(''T''*''M'') &rarr; ''TM''}} and applying &omega; to this projection.  Note that the tautological one-form is not a pullback of a one-form on the base ''M''.

=== Symplectic form ===

The cotangent bundle has a canonical [[symplectic form|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 |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 }}

{{Manifolds}}

[[Category:Vector bundles]]
[[Category:Differential topology]]
[[Category:Tensors]]