Editing Cotangent bundle (section)

== 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 &theta; called the [[tautological one-form]], discussed below. The [[exterior derivative]] of &theta; is a [[symplectic form|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 manifold]]s, any real function on the cotangent bundle can be interpreted to be a [[symplectic vector space|Hamiltonian]]; thus the cotangent bundle can be understood to be a [[phase space]] on which [[Hamiltonian mechanics]] plays out.

=== The tautological one-form ===
{{main|Tautological one-form}}

The cotangent bundle carries a canonical one-form &theta; 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 (fiber bundle)|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''<sup>''i''</sup> are local coordinates on the base manifold ''M''.  In terms of these base coordinates, there are fibre coordinates ''p''<sub>''i''</sub> :  a one-form at a particular point of ''T''*''M'' has the form ''p''<sub>''i''</sub>&nbsp;''dx''<sup>''i''</sup> ([[Einstein summation convention]] implied).  So the manifold ''T''*''M'' itself carries local coordinates (''x''<sup>''i''</sup>, ''p''<sub>''i''</sub>) where the ''x''<nowiki/>'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

:<math>\theta_{(x,p)}=\sum_{i=1}^n p_i \, dx^i.</math>

Intrinsically, the value of the canonical one-form in each fixed point of ''T*M'' is given as a [[pullback (differential geometry)|pullback]].  Specifically, suppose that {{nowrap|&pi; : ''T*M'' &rarr; ''M''}} is the [[Projection (mathematics)|projection]] of the bundle. Taking a point in ''T''<sub>''x''</sub>*''M'' is the same as choosing of a point ''x'' in ''M'' and a one-form &omega; at ''x'', and the tautological one-form &theta; assigns to the point (''x'', &omega;) the value

:<math>\theta_{(x,\omega)}=\pi^*\omega.</math>

That is, for a vector ''v'' in the tangent bundle of the cotangent bundle, the application of the tautological one-form &theta; to ''v'' at (''x'', &omega;) is computed by projecting ''v'' into the tangent bundle at ''x'' using {{nowrap|''d''&pi; : ''T''(''T''*''M'') &rarr; ''TM''}} and applying &omega; 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 form|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 <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 \land 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 ''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.