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.
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 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:
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 the vanishing locus of a smooth function , its tangent bundle is:
where is the covector defined by the directional derivative . Its cotangent bundle consists of pairs , where and we take the covector in the quotient space of modulo the line generated by . Of course, the dot product identifies the quotient space with the orthogonal space to the gradient , so the two bundles are isomorphic.
For example, let M be the 3-sphere given by the vanishing locus of in . Its tangent bundle is the set of with and . Its cotangent bundle is the set of with the covector considered modulo the direction ; so at , we take modulo .
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, 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 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
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*M → M 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
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.
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 .
If the manifold 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 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.
- 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.