Cotangent bundle
In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold.This may be generalized to categories with more structure than smooth manifolds, such as complex manifolds, or (in the form of cotangent sheaf) algebraic varieties or 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.There are several equivalent ways to define the cotangent bundle.One way is through a diagonal mapping Δ and germs.be the sheaf of germs of smooth functions on M×M which vanish on the diagonal.consists of equivalence classes of functions which vanish on the diagonal modulo higher order terms.Smooth sections of the cotangent bundle are called (differential) one-forms.denotes the dual space of covectors, linear functionsembedded as a hypersurface represented by the vanishing locus of a functionBecause at each point the tangent directions of M can be paired with their dual covectors in the fiber, X possesses a canonical one-form θ called the tautological one-form, discussed below.The exterior derivative of θ is a symplectic 2-form, out of which a non-degenerate volume form can be built for X.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 cotangent bundle carries a canonical one-form θ 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 of the vector bundle T*(T*M) over T*M. This section can be constructed in several ways.Suppose that xi are local coordinates on the base manifold M. In terms of these base coordinates, there are fibre coordinates pi : a one-form at a particular point of T*M has the form pi dxi (Einstein summation convention implied).Specifically, suppose that π : T*M → M is the projection of the bundle.Taking a point in Tx*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 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 dπ : T(T*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. 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, and the differential is the canonical symplectic form, the sum ofrepresents the set of possible positions in a dynamical system, then the cotangent bundleFor 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.