Editing Poisson bracket (section)

==Definition in canonical coordinates==
In [[canonical coordinates]] (also known as [[Darboux coordinates]]) <math> (q_i,\, p_i)</math> on the [[phase space]], given two functions <math> f(p_i,\, q_i, t)</math> and <math> g(p_i,\, q_i, t)</math>,<ref group="Note"><math> f(p_i,\, q_i,\, t)</math> means <math>f</math> is a function of the <math>2N + 1</math> independent variables: momentum, <math>p_{1 \dots N}</math>; position, <math>q_{1 \dots N}</math>; and time, <math>t</math></ref> the Poisson bracket takes the form
<math display="block">\{f, g\} = \sum_{i=1}^{N} \left( \frac{\partial f}{\partial q_{i}} \frac{\partial g}{\partial p_{i}} - \frac{\partial f}{\partial p_i} \frac{\partial g}{\partial q_i}\right).</math>

The Poisson brackets of the canonical coordinates are
<math display="block">\begin{align}
  \{q_k,q_l\} &= \sum_{i=1}^{N} \left( \frac{\partial q_k}{\partial q_{i}} \frac{\partial q_l}{\partial p_{i}} - \frac{\partial q_k}{\partial p_i} \frac{\partial q_l}{\partial q_i}\right) =  \sum_{i=1}^{N} \left( \delta_{ki} \cdot 0 - 0 \cdot \delta_{li}\right) = 0, \\
  \{p_k,p_l\} &=\sum_{i=1}^{N} \left( \frac{\partial p_k}{\partial q_{i}} \frac{\partial p_l}{\partial p_{i}} - \frac{\partial p_k}{\partial p_i} \frac{\partial p_l}{\partial q_i}\right)  =  \sum_{i=1}^{N} \left( 0 \cdot \delta_{li} - \delta_{ki} \cdot 0\right) =  0, \\
  \{q_k,p_l\} &= \sum_{i=1}^{N} \left( \frac{\partial q_k}{\partial q_{i}} \frac{\partial p_l}{\partial p_{i}} - \frac{\partial q_k}{\partial p_i} \frac{\partial p_l}{\partial q_i}\right) =  \sum_{i=1}^{N} \left( \delta_{ki} \cdot \delta_{li} - 0 \cdot 0\right) = \delta_{kl},
\end{align}</math>
where <math>\delta_{ij}</math> is the [[Kronecker delta]].