5.1 Canonical Bracket
For a smooth function \(F\) on phase space, write \(\partial _{q_i}F\) for its partial derivative in the \(q_i\) direction and \(\partial _{p_i}F\) in the \(p_i\) direction. The Poisson bracket is:
\[ \{ F, G\} = \sum _{i=1}^3 \! \left(\partial _{q_i}F \cdot \partial _{p_i}G - \partial _{p_i}F \cdot \partial _{q_i}G\right). \]
If \(H\) is the Hamiltonian, then \(\{ F,H\} \) is the time derivative of \(F\) along the motion.
The coordinate functions satisfy:
\[ \{ q_i, q_j\} = 0, \qquad \{ p_i, p_j\} = 0, \qquad \{ q_i, p_j\} = \delta _{ij}. \]
Proof
From the definition, \(\{ q_i, q_j\} = \sum _k(\delta _{ik} \cdot 0 - 0 \cdot \delta _{jk}) = 0\), and \(\{ p_i, p_j\} = 0\) similarly. For the mixed bracket: \(\{ q_i, p_j\} = \sum _k \delta _{ik}\delta _{jk} = \delta _{ij}\).
The Kepler equations of motion arise from the Poisson bracket with \(H\):
\[ \{ q_i, H\} = \frac{p_i}{m}, \qquad \{ p_i, H\} = -\frac{\mu q_i}{\| q\| ^3}. \]
Proof
By the canonical brackets and the chain rule:
\[ \{ q_i, H\} = \frac{\partial H}{\partial p_i} = \frac{p_i}{m}. \]
For the momentum bracket:
\[ \{ p_i, H\} = -\frac{\partial H}{\partial q_i} = -\frac{\partial }{\partial q_i}\! \left(-\frac{\mu }{\| q\| }\right) = -\frac{\mu q_i}{\| q\| ^3}. \]