Cotangent bundle

Last updated

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.

Formal Definition

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 ${\displaystyle {\mathcal {I}}}$ be the sheaf of germs of smooth functions on M×M which vanish on the diagonal. Then the quotient sheaf ${\displaystyle {\mathcal {I}}/{\mathcal {I}}^{2}}$ consists of equivalence classes of functions which vanish on the diagonal modulo higher order terms. The cotangent sheaf is defined as the pullback of this sheaf to M:

${\displaystyle \Gamma T^{*}M=\Delta ^{*}\left({\mathcal {I}}/{\mathcal {I}}^{2}\right).}$

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.

Contravariance Properties

A smooth morphism ${\displaystyle \phi \colon M\to N}$ of manifolds, induces a pullback sheaf ${\displaystyle \phi ^{*}T^{*}N}$ on M. There is an induced map of vector bundles ${\displaystyle \phi ^{*}(T^{*}N)\to T^{*}M}$.

Examples

The tangent bundle of the vector space ${\displaystyle \mathbb {R} ^{n}}$ is ${\displaystyle T\,\mathbb {R} ^{n}=\mathbb {R} ^{n}\times \mathbb {R} ^{n}}$, and the cotangent bundle is ${\displaystyle T^{*}\mathbb {R} ^{n}=\mathbb {R} ^{n}\times (\mathbb {R} ^{n})^{*}}$, where ${\displaystyle (\mathbb {R} ^{n})^{*}}$ denotes the dual space of covectors, linear functions ${\displaystyle v^{*}:\mathbb {R} ^{n}\to \mathbb {R} }$.

Given a smooth manifold ${\displaystyle M\subset \mathbb {R} ^{n}}$ embedded as a hypersurface represented by the vanishing locus of a function ${\displaystyle f\in C^{\infty }(\mathbb {R} ^{n}),}$ with the condition that ${\displaystyle \nabla f\neq 0,}$ the tangent bundle is

${\displaystyle TM=\{(x,v)\in T\,\mathbb {R} ^{n}\$ :\ f(x)=0,\ \,df_{x}(v)=0\},}

where ${\displaystyle df_{x}\in T_{x}^{*}M}$ is the directional derivative ${\displaystyle df_{x}(v)=\nabla \!f(x)\cdot v}$. By definition, the cotangent bundle in this case is

${\displaystyle T^{*}M={\bigl \{}(x,v^{*})\in T^{*}\mathbb {R} ^{n}\$ :\ f(x)=0,\ v^{*}\in T_{x}^{*}M{\bigr \}},}

where ${\displaystyle T_{x}^{*}M=\{v\in T_{x}\mathbb {R} ^{n}\$ :\ df_{x}(v)=0\}^{*}.} Since every covector ${\displaystyle v^{*}\in T_{x}^{*}M}$ corresponds to a unique vector ${\displaystyle v\in T_{x}M}$ for which ${\displaystyle v^{*}(u)=v\cdot u,}$ for an arbitrary ${\displaystyle u\in T_{x}M,}$

${\displaystyle 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 \}}.}

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 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 tautological one-form

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

${\displaystyle \theta _{(x,p)}=\sum _{{\mathfrak {i}}=1}^{n}p_{i}\,dx^{i}.}$

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*MM 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

${\displaystyle \theta _{(x,\omega )}=\pi ^{*}\omega .}$

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.

Symplectic form

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 ${\displaystyle \mathbb {R} ^{n}\times \mathbb {R} ^{n}}$. But there the one form defined is the sum of ${\displaystyle y_{i}\,dx_{i}}$, and the differential is the canonical symplectic form, the sum of ${\displaystyle dy_{i}\land dx_{i}}$.

Phase space

If the manifold ${\displaystyle M}$ represents the set of possible positions in a dynamical system, then the cotangent bundle ${\displaystyle \!\,T^{*}\!M}$ 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 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.

Related Research Articles

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.

In differential geometry, the tangent bundle of a differentiable manifold is a manifold which assembles all the tangent vectors in . As a set, it is given by the disjoint union of the tangent spaces of . That is,

In mathematics, a symplectic vector space is a vector space V over a field F equipped with a symplectic bilinear form.

In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles. They have since found applications in physics, Calabi–Yau manifolds, string theory, Chern–Simons theory, knot theory, Gromov–Witten invariants, topological quantum field theory, the Chern theorem etc.

In mathematics, contact geometry is the study of a geometric structure on smooth manifolds given by a hyperplane distribution in the tangent bundle satisfying a condition called 'complete non-integrability'. Equivalently, such a distribution may be given as the kernel of a differential one-form, and the non-integrability condition translates into a maximal non-degeneracy condition on the form. These conditions are opposite to two equivalent conditions for 'complete integrability' of a hyperplane distribution, i.e. that it be tangent to a codimension one foliation on the manifold, whose equivalence is the content of the Frobenius theorem.

In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with reference to a sheaf of rings that codifies this geometric information.

In physics and mathematics, supermanifolds are generalizations of the manifold concept based on ideas coming from supersymmetry. Several definitions are in use, some of which are described below.

In differential geometry, a G-structure on an n-manifold M, for a given structure group G, is a principal G-subbundle of the tangent frame bundle FM of M.

In mathematics and physics, a Hamiltonian vector field on a symplectic manifold is a vector field, defined for any energy function or Hamiltonian. Named after the physicist and mathematician Sir William Rowan Hamilton, a Hamiltonian vector field is a geometric manifestation of Hamilton's equations in classical mechanics. The integral curves of a Hamiltonian vector field represent solutions to the equations of motion in the Hamiltonian form. The diffeomorphisms of a symplectic manifold arising from the flow of a Hamiltonian vector field are known as canonical transformations in physics and (Hamiltonian) symplectomorphisms in mathematics.

In mathematics, the tautological one-form is a special 1-form defined on the cotangent bundle of a manifold . In physics, it is used to create a correspondence between the velocity of a point in a mechanical system and its momentum, thus providing a bridge between Lagrangian mechanics with Hamiltonian mechanics.

In mathematics, a differentiable manifold is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since each chart lies within a linear space to which the usual rules of calculus apply. If the charts are suitably compatible, then computations done in one chart are valid in any other differentiable chart.

In differential geometry, a field of mathematics, a normal bundle is a particular kind of vector bundle, complementary to the tangent bundle, and coming from an embedding.

In mathematics, specifically in symplectic geometry, the momentum map is a tool associated with a Hamiltonian action of a Lie group on a symplectic manifold, used to construct conserved quantities for the action. The momentum map generalizes the classical notions of linear and angular momentum. It is an essential ingredient in various constructions of symplectic manifolds, including symplectic (Marsden–Weinstein) quotients, discussed below, and symplectic cuts and sums.

In mathematics, a vector-valued differential form on a manifold M is a differential form on M with values in a vector space V. More generally, it is a differential form with values in some vector bundle E over M. Ordinary differential forms can be viewed as R-valued differential forms.

In mathematics, a holomorphic vector bundle is a complex vector bundle over a complex manifold X such that the total space E is a complex manifold and the projection map π : EX is holomorphic. Fundamental examples are the holomorphic tangent bundle of a complex manifold, and its dual, the holomorphic cotangent bundle. A holomorphic line bundle is a rank one holomorphic vector bundle.

In the field of mathematics known as differential geometry, a generalized complex structure is a property of a differential manifold that includes as special cases a complex structure and a symplectic structure. Generalized complex structures were introduced by Nigel Hitchin in 2002 and further developed by his students Marco Gualtieri and Gil Cavalcanti.

In mathematics, more precisely in differential geometry, a soldering of a fiber bundle to a smooth manifold is a manner of attaching the fibers to the manifold in such a way that they can be regarded as tangent. Intuitively, soldering expresses in abstract terms the idea that a manifold may have a point of contact with a certain model Klein geometry at each point. In extrinsic differential geometry, the soldering is simply expressed by the tangency of the model space to the manifold. In intrinsic geometry, other techniques are needed to express it. Soldering was introduced in this general form by Charles Ehresmann in 1950.

In algebraic geometry, given a morphism f: XS of schemes, the cotangent sheaf on X is the sheaf of -modules that represents S-derivations in the sense: for any -modules F, there is an isomorphism

References

• Abraham, Ralph; Marsden, Jerrold E. (1978). Foundations of Mechanics. London: Benjamin-Cummings. ISBN   0-8053-0102-X.
• Jost, Jürgen (2002). Riemannian Geometry and Geometric Analysis. Berlin: Springer-Verlag. ISBN   3-540-63654-4.
• Singer, Stephanie Frank (2001). Symmetry in Mechanics: A Gentle Modern Introduction. Boston: Birkhäuser.