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. 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 (some? all?) 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.
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One-forms
Smooth sections of the cotangent bundle are differential one-forms.
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 <math>\mathbb{R}^n \times \mathbb{R}^n<math>. But there the one form defined is the sum of <math>y_{i}dx_i<math>, and the differential is the canonical symplectic form, the sum of <math>dy_i{\and}dx_i<math>.
Phase space
If the manifold <math>M<math> represents the set of possible positions in a dynamical system, then the cotangent bundle <math>\!\,T^{*}\!M<math> can be thought of as the set of possible positions and momentums. For example, this is an easy way to describe the (non-trivial) phase space of a three-dimensional spherical pendulum: a weighted ball constrained to move along a 2-sphere. 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.
Related topics
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.