In differential geometry, the cotangent bundle of a manifold is the vector bundle of all the cotangent spaces at every point in the manifold. Cotangent spaces possess a canonical symplectic 2-form out of which a non-degenerate volume form can be built for the cotangent bundle. As a result, the cotangent bundle, considered as a manifold itself, is always orientable. 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. In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ... This is the current mathematics collaboration of the week! Please help improve it to featured article standard at manifold/rewrite. ... In mathematics, a vector bundle is a geometrical construct where to every point of a topological space (or manifold, or algebraic variety) we attach a vector space in a compatible way, so that all those vector spaces, glued together, form another topological space (or manifold or variety). ... In differential geometry, one can attach to every point p of a differentiable manifold a vector space called the cotangent space at p. ... In mathematics, a symplectic manifold is a smooth manifold equipped with a closed, nondegenerate 2-form. ... In mathematics, the volume form is a differential form that represents a unit volume of a Riemannian manifold or a pseudo-Riemannian manifold. ... This article or section should be merged with Orientable manifold. ... See Cartesian coordinate system or Coordinates (elementary mathematics) for a more elementary introduction to this topic. ... In physics, the canonical commutation relation is the relation among the position and momentum of a point particle in one dimension, where is the so-called commutator of and , is the imaginary unit and is the reduced Plancks constant. ... In mathematics, a symplectic manifold is a smooth manifold equipped with a closed, nondegenerate 2-form. ... In mathematics, a symplectic vector space is a vector space V equipped with a nondegenerate, skew-symmetric, bilinear form ω called the symplectic form. ... Phase space is a useful construct in mathematics and physics to demonstrate and visualise the changes in the dynamical variables of a system. ... Hamiltonian mechanics is a re-formulation of classical mechanics that was invented in 1833 by William Rowan Hamilton. ...

One-forms

Smooth sections of the cotangent bundle are differential one-forms. In mathematics, a smooth function is one that is infinitely differentiable, i. ... In mathematics, in particular in topology, a fiber bundle is a space which locally looks like a product of two spaces but may possess a different global structure. ... A one-form is a linear function which maps each vector in a vector space to a real number, such that the mapping is invariant with respect to coordinate transformations of the vector space. ...

The cotangent bundle as phase space

Remark: This article needs to clarify the difference between locally Hamiltonian systems and globally Hamiltonian systems, and specifically provide examples where a cotangent bundle cannot be thought of as a phase space of a dynamical system (at least globally).

Symplectic form

The cotangent bundle has a canonical symplectic 2-form on it, as an exterior derivative of a one-form. 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 . But there the one form defined is the sum of y_idx_i, and the differential is the canonical symplectic form, the sum of . In mathematics, a symplectic manifold is a smooth manifold equipped with a closed, nondegenerate 2-form. ... In mathematics, the exterior derivative operator of differential topology, extends the concept of the differential of a function to differential forms of higher degree. ... A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ...

Phase space

If the manifold M 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 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. A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. ... Phase space is a useful construct in mathematics and physics to demonstrate and visualise the changes in the dynamical variables of a system. ... Hamiltonian mechanics is a re-formulation of classical mechanics that was invented in 1833 by William Rowan Hamilton. ... This is a glossary of some terms used in Riemannian geometry and metric geometry — it doesnt cover the terminology of differential topology. ...

Related topics

• Tangent bundle

In mathematics, the tangent bundle of a differentiable manifold is a vector bundle which as a set is the disjoint union of all the tangent spaces at every point in the manifold with natural topology and smooth structure. ...

References

• Jurgen Jost, Riemannian Geometry and Geometric Analysis, (2002) Springer-Verlag, Berlin ISBN 3-540-4267-2.
• Ralph Abraham and Jarrold 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.