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| 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]].
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| == The cotangent sheaf ==
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| [[Smooth function|Smooth]] [[Fiber bundle|sections]] of the cotangent bundle are differential [[one-form]]s.
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| === Definition of the cotangent sheaf ===
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| Let ''M'' be a [[Differentiable manifold|smooth manifold]] and let ''M''×''M'' be the [[Cartesian product]] of ''M'' with itself. The [[diagonal mapping]] Δ sends a point ''p'' in ''M'' to the point (''p'',''p'') of ''M''×''M''. The image of Δ is called the diagonal. Let <math>\mathcal{I}</math> be the [[sheaf (mathematics)|sheaf]] of [[germ (mathematics)|germs]] of smooth functions on ''M''×''M'' which vanish on the diagonal. Then the 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 the [[inverse image functor|pullback]] of this sheaf to ''M'':
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| :<math>\Gamma T^*M=\Delta^*(\mathcal{I}/\mathcal{I}^2).</math>
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| 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'''.
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| === Contravariance in manifolds ===
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| 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>.
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| == The cotangent bundle as phase space ==
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| Since the cotangent bundle ''X''=''T''*''M'' is a [[vector bundle]], it can be regarded as a manifold in its own right. Because of the manner in which the definition of ''T''*''M'' relates to the [[differential topology]] of the base space ''M'', ''X'' possesses a canonical one-form θ (also [[tautological one-form]] or [[symplectic potential]]). The [[exterior derivative]] of θ 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 (meaning that the tangent bundle of ''X'' 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.
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| === The tautological one-form ===
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| {{main|Tautological one-form}}
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| The cotangent bundle carries a tautological one-form θ also known as the ''Poincaré'' ''1''-form or ''Liouville'' ''1''-form. (The form is also known as the ''canonical one-form'', although this can sometimes lead to confusion.) 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''.
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| This section can be constructed in several ways. The most elementary method is to use 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>''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'' are coordinates on the base and the ''p'' are coordinates in the fibre. The canonical one-form is given in these coordinates by
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| :<math>\theta_{(x,p)}=\sum_{{\mathfrak i}=1}^n p_idx^i.</math>
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| 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|π : ''T*M'' → ''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 ω at ''x'', and the tautological one-form θ assigns to the point (''x'', ω) the value
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| :<math>\theta_{(x,\omega)}=\pi^*\omega.</math>
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| That is, for a vector ''v'' in the tangent bundle of the cotangent bundle, the application of the tautological one-form θ to ''v'' at (''x'', ω) is computed by projecting ''v'' into the tangent bundle at ''x'' using {{nowrap|''d''π : ''TT''*''M'' → ''TM''}} and applying ω to this projection. Note that the tautological one-form is not a pullback of a one-form on the base ''M''.
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| === Symplectic form ===
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| The cotangent bundle has a canonical [[symplectic form|symplectic 2-form]] on it, as an [[exterior derivative]] of the [[tautological one-form|canonical 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{\and}dx_i</math>.
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| === Phase space ===
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| 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 not changing). The entire state space ''looks like'' a cylinder. The cylinder 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]] for more information, and the article on [[geodesic flow]] for an explicit construction of the Hamiltonian equations of motion.
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| ==See also==
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| * [[Legendre transformation]]
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| ==References==
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| * Jurgen Jost, ''Riemannian Geometry and Geometric Analysis'', (2002) Springer-Verlag, Berlin ISBN 3-540-63654-4.
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| * [[Ralph Abraham]] and [[Jerrold E. Marsden]], ''Foundations of Mechanics'', (1978) Benjamin-Cummings, London ISBN 0-8053-0102-X.
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| * Stephanie Frank Singer, ''Symmetry in Mechanics: A Gentle Modern Introduction'', (2001) Birkhauser, Boston.
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| [[Category:Vector bundles]]
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| [[Category:Differential topology]]
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