Exercise 1

\[\{ F(q, p), p_i\} = \sum_j \frac{\partial F}{\partial q_j} \frac{\partial p_i}{\partial p_j} - \frac{\partial F}{\partial p_j} \frac{\partial p_i}{\partial q_j}\]

Since \[\frac{\partial p_i}{\partial q_j} = 0\] and \[\frac{\partial p_i}{\partial p_j} = \delta_{ij}\] it follows: \[\{ F(q, p), p_i\} = \sum_j \frac{\partial F}{\partial q_j} \delta_{ij} = \frac{\partial F}{\partial q_i}\]

\[\{ F(q, p), q_i\} = \sum_j \frac{\partial F}{\partial q_j} \frac{\partial q_i}{\partial p_j} - \frac{\partial F}{\partial p_j} \frac{\partial q_i}{\partial q_j}\]

Since \[\frac{\partial q_i}{\partial p_j} = 0\] and \[\frac{\partial q_i}{\partial q_j} = \delta_{ij}\] it follows: \[\{ F(q, p), q_i\} = - \sum_j \frac{\partial F}{\partial p_j} \delta_{ij} = - \frac{\partial F}{\partial p_i}\]