User:Nik

Let \( \Phi \in C^\infty(\mathbb{R}^n) \) have the property that the system \( p_i = \frac{\partial}{\partial q_i} \Phi (q, \widetilde{q}) \) has a unique smooth solution \( \widetilde{q} = \widetilde{q} (q,p) \).

Define \( \widetilde{p_i}(q,p) = - \frac{\partial}{\partial \widetilde{q_i}} \Phi (q, \widetilde{q}) | _{\widetilde{q}= \widetilde{q}(q,p)} \)

Let \( \{\cdot,\cdot\} \) be the Poisson bracket, such that \( \{f,g\} = \sum_{j=1}^n \frac{\partial f}{\partial q_j} \frac{\partial g}{\partial p_j} - \frac{\partial f}{\partial p_j} \frac{\partial g}{\partial q_j} \)

Show that:

i) \( \{\widetilde{q_i} (q,p), \widetilde{q_j} (q,p)\} = \{\widetilde{p_i} (q,p), \widetilde{p_j} (q,p)\} = 0 \)

ii) \( \{\widetilde{q_i} (q,p), \widetilde{p_j} (q,p)\} = \delta_{ij} \)