Editing Poisson bracket (section)

==A result on conjugate momenta==
Given a smooth [[vector field]] <math>X</math> on the configuration space, let <math>P_X</math> be its [[conjugate momentum]]. The conjugate momentum mapping is a [[Lie algebra]] anti-homomorphism from the [[Lie bracket of vector fields|Lie bracket]] to the Poisson bracket:
<math display="block">\{P_X, P_Y\} = -P_{[X, Y]}.</math>

This important result is worth a short proof. Write a vector field <math>X</math> at point <math>q</math> in the [[Configuration space (physics)|configuration space]] as
<math display="block">X_q = \sum_i X^i(q) \frac{\partial}{\partial q^i}</math>
where <math display="inline"> \frac{\partial}{\partial q^i}</math> is the local coordinate frame. The conjugate momentum to <math>X</math> has the expression
<math display="block">P_X(q, p) = \sum_i X^i(q) \;p_i</math>
where the <math>p_i</math> are the momentum functions conjugate to the coordinates. One then has, for a point <math>(q,p)</math> in the [[phase space]],
<math display="block">\begin{align}
  \{P_X,P_Y\}(q,p) &=  \sum_i \sum_j \left\{ X^i(q) \;p_i, Y^j(q)\; p_j \right\} \\
                   &=  \sum_{ij} p_i Y^j(q) \frac{\partial X^i}{\partial q^j} -  p_j X^i(q) \frac{\partial Y^j}{\partial q^i} \\
                   &= -\sum_i p_i \; [X, Y]^i(q) \\
                   &= - P_{[X, Y]}(q, p). 
\end{align}</math>

The above holds for all <math>(q, p)</math>, giving the desired result.