In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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
be the
sheaf of
germs of smooth functions on ''M''×''M'' which vanish on the diagonal. Then the
quotient sheaf
In mathematics, a sheaf is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the data could ...
consists of equivalence classes of functions which vanish on the diagonal modulo higher order terms. The
cotangent sheaf In algebraic geometry, given a morphism ''f'': ''X'' → ''S'' of schemes, the cotangent sheaf on ''X'' is the sheaf of \mathcal_X-modules \Omega_ that represents (or classifies) ''S''- derivations in the sense: for any \mathcal_X-modules ''F'', t ...
is defined as the
pullback
In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward.
Precomposition
Precomposition with a function probably provides the most elementary notion of 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
Section, Sectioning or Sectioned may refer to:
Arts, entertainment and media
* Section (music), a complete, but not independent, musical idea
* Section (typography), a subdivision, especially of a chapter, in books and documents
** Section sig ...
of the cotangent bundle are called (differential)
one-forms.
Contravariance Properties
A smooth morphism
of manifolds, induces a
pullback sheaf on ''M''. There is an
induced map of vector bundles
.
Examples
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
:
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
In mathematics and classical mechanics, canonical coordinates are sets of coordinates on phase space which can be used to describe a physical system at any given point in time. Canonical coordinates are used in the Hamiltonian formulation of cl ...
. 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, ''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. The most elementary method uses local coordinates. Suppose that ''x''
''i'' are local coordinates on the base manifold ''M''. In terms of these base coordinates, there are fibre coordinates ''p''
''i'': a one-form at a particular point of ''T''*''M'' has the form ''p''
''i'' ''dx''
''i'' (
Einstein summation convention implied). So the manifold ''T''*''M'' itself carries local coordinates (''x''
''i'', ''p''
''i'') 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
In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward.
Precomposition
Precomposition with a function probably provides the most elementary notion of pullback: ...
. Specifically, suppose that is the
projection of the bundle. Taking a point in ''T''
''x''*''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 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
. But there the one form defined is the sum of
, and the differential is the canonical symplectic form, the sum of
.
Phase space
If the manifold
represents the set of possible positions in a
dynamical system
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water i ...
, 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
In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of ...
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.
See also
*
Legendre transformation
References
*
*
*
{{Manifolds
Vector bundles
Differential topology
Tensors