Cotangent bundle

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In differential geometry, the cotangent bundle of a manifold is the vector bundle of all the cotangent spaces at every point in the manifold.

One-forms (the cotangent sheaf)

Smooth sections of the cotangent bundle are differential one-forms.

Definition of the cotangent sheaf

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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{I}} be the sheaf of germs of smooth functions on M×M which vanish on the diagonal. Then the quotient sheaf Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{I}/\mathcal{I}^2} consists of equivalence classes of functions which vanish on the diagonal modulo higher order terms. The cotangent sheaf is the pullback of this sheaf to M:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma T^*M=\Delta^*(\mathcal{I}/\mathcal{I}^2).}

By Taylor's theorem, this is a locally finite 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.

The cotangent bundle as phase space

The cotangent bundle X=T*M, since it is a vector bundle, can be regarded as a manifold in its own right. Because of the manner in which the defintion of T*M relates to the differential topology of the base space M, X possess a tautological one-form θ (the symplectic potential). 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 (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 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 canonical one-form

The cotangent bundle carries a tautological one-form θ (usually called 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 of the vector bundle T*(T*M) over T*M. This section can be constructed in several ways. The most elementary method is to use 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 pidxi (Einstein summation convention implied). So the manifold T*M itself caries local coordinates (xi,pi) 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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta_{(x,p)}=\sum_{{\mathfrak i}=1}^n p_idx^i}

Intrinsically, the canonical one-form is given as a pullback. Specifically, if π:T*MM is the projection of the bundle, then if ω is a one-form at a point x of M (hence an element of T*M),

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta_{(x,\omega)}=\pi^*\omega.}

Symplectic form

The cotangent bundle has a canonical symplectic 2-form on it, as an exterior derivative of a one-form, the symplectic potential. The one-form assigns to a vector in the tangent bundle of the cotangent bundle the application of the element in the cotangent bundle (a linear functional) to the projection of the vector into the tangent bundle (the differential of the projection of the cotangent bundle to the original manifold). Proving 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}^n \times \mathbb{R}^n} . But there the one form defined is the sum of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_{i}dx_i} , and the differential is the canonical symplectic form, the sum of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle dy_i{\and}dx_i} .

See main article tautological one-form for details.

Phase space

If the manifold Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} represents the set of possible positions in a dynamical system, then the cotangent bundle Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \!\,T^{*}\!M} 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.

See also

References

  • Jurgen Jost, Riemannian Geometry and Geometric Analysis, (2002) Springer-Verlag, Berlin ISBN 3-540-4267-2.
  • Ralph Abraham and Jerrold E. Marsden, Foundations of Mechanics, (1978) Benjamin-Cummings, London ISBN 0-8053-0102-X.
  • Stephanie Frank Singer, Symmetry in Mechanics: A Gentle Modern Introduction, (2001) Birkhauser, Boston.

es:Fibrado cotangente zh:余切丛