Editing Canonical coordinates (section)

==Formal development==
Given a manifold {{mvar|Q}}, a [[vector field]] {{mvar|X}} on {{mvar|Q}} (a [[section (fiber bundle)|section]] of the [[tangent bundle]] {{math|''TQ''}}) can be thought of as a function acting on the [[cotangent bundle]], by the duality between the tangent and cotangent spaces.  That is, define a function

:<math>P_X: T^*Q \to \mathbb{R}</math>

such that

:<math>P_X(q, p) = p(X_q)</math>

holds for all cotangent vectors {{mvar|p}} in <math>T_q^*Q</math>. Here, <math>X_q</math> is a vector in <math>T_qQ</math>, the tangent space to the manifold {{mvar|Q}} at point {{mvar|q}}.  The function <math>P_X</math> is called the ''momentum function'' corresponding to {{mvar|X}}.

In [[atlas (topology)|local coordinates]], the vector field {{mvar|X}} at point {{mvar|q}} may be written as

:<math>X_q = \sum_i X^i(q) \frac{\partial}{\partial q^i}</math>

where the <math>\partial /\partial q^i</math> are the coordinate frame on {{mvar|TQ}}. The conjugate momentum then has the expression

:<math>P_X(q, p) = \sum_i X^i(q)\; p_i</math>

where the <math>p_i</math> are defined as the momentum functions corresponding to the vectors <math>\partial /\partial q^i</math>:

:<math>p_i = P_{\partial /\partial q^i}</math>

The <math>q^i</math> together with the <math>p_j</math> together form a coordinate system on the cotangent bundle <math>T^*Q</math>; these coordinates are called the ''canonical coordinates''.