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. It may be described also as the dual bundle to the tangent bundle. 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.
Let M be a 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 be the sheaf of germs of smooth functions on M×M which vanish on the diagonal. Then the quotient sheaf consists of equivalence classes of functions which vanish on the diagonal modulo higher order terms. The cotangent sheaf is defined as the pullback of this sheaf to M:
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 sections of the cotangent bundle are called (differential) one-forms.
A smooth morphism of manifolds, induces a pullback sheaf on M. There is an induced map of vector bundles .
The tangent bundle of the vector space is , and the cotangent bundle is , where denotes the dual space of covectors, linear functions .
Given a smooth manifold embedded as a hypersurface represented by the vanishing locus of a function with the condition that the tangent bundle is
where is the directional derivative . By definition, the cotangent bundle in this case is
where Since every covector corresponds to a unique vector for which for an arbitrary
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 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. 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 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, Poincaré1-form, or Liouville1-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. The most elementary method uses local coordinates. 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). So the manifold T*M itself carries local coordinates (xi, pi) where the x'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
Intrinsically, the value of the canonical one-form in each fixed point of T*M is given as a pullback. 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 . But there the one form defined is the sum of , and the differential is the canonical symplectic form, the sum of .
If the manifold represents the set of possible positions in a dynamical system, then the cotangent bundle 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.
In differential geometry, one can attach to every point of a smooth manifold, , a vector space called the cotangent space at . Typically, the cotangent space, is defined as the dual space of the tangent space at , , although there are more direct definitions. The elements of the cotangent space are called cotangent vectors or tangent covectors.
In differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, , equipped with a closed nondegenerate differential 2-form , called the symplectic form. The study of symplectic manifolds is called symplectic geometry or symplectic topology. Symplectic manifolds arise naturally in abstract formulations of classical mechanics and analytical mechanics as the cotangent bundles of manifolds. For example, in the Hamiltonian formulation of classical mechanics, which provides one of the major motivations for the field, the set of all possible configurations of a system is modeled as a manifold, and this manifold's cotangent bundle describes the phase space of the system.
Hamiltonian mechanics is a mathematically sophisticated formulation of classical mechanics. Historically, it contributed to the formulation of statistical mechanics and quantum mechanics. Hamiltonian mechanics was first formulated by William Rowan Hamilton in 1833, starting from Lagrangian mechanics, a previous reformulation of classical mechanics introduced by Joseph Louis Lagrange in 1788. Like Lagrangian mechanics, Hamiltonian mechanics is equivalent to Newton's laws of motion in the framework of classical mechanics.
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