Editing Symplectic manifold (section)

=== Cotangent bundles ===
Let <math>Q</math> be a smooth manifold of dimension <math>n</math>. Then the total space of the [[cotangent bundle]] <math>T^* Q</math> has a natural symplectic form, called the Poincaré two-form or the [[canonical symplectic form]]

:<math>\omega = \sum_{i=1}^n dp_i \wedge dq^i </math>

Here <math>(q^1, \ldots, q^n)</math> are any local coordinates on <math>Q</math> and <math>(p_1, \ldots, p_n)</math> are fibrewise coordinates with respect to the cotangent vectors <math>dq^1, \ldots, dq^n</math>. Cotangent bundles are the natural [[phase space]]s of classical mechanics.  The point of distinguishing upper and lower indexes is driven by the case of the manifold having a [[metric tensor]], as is the case for [[Riemannian manifold]]s. Upper and lower indexes transform contra and covariantly under a change of coordinate frames.  The phrase "fibrewise coordinates with respect to the cotangent vectors" is meant to convey that the momenta <math>p_i</math> are "[[solder form|soldered]]" to the velocities <math>dq^i</math>. The soldering is an expression of the idea that velocity and momentum are colinear, in that both move in the same direction, and differ by a scale factor.