User:Nik
From Ferienserie MMP2
Revision as of 10:55, 30 December 2014 by Un vieil homme (Talk | contribs)
Foreword
I use \(Q\:/\:P\) instead of \(\widetilde{q}\:/\:\widetilde{p}\) because it's easier to write in Latex.
Problem
Let \( \Phi \in C^\infty(\mathbb{R}^n) \) have the property that the system \( p_i = \frac{\partial}{\partial q_i} \Phi (q, Q \) has a unique smooth solution \( Q = Q(q,p) \).
Define \( P_i(q,p) = - \frac{\partial}{\partial Q_i} \Phi (q, Q) | _{Q= Q(q,p)} \)
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) \( \{Q_i(q,p), Q_j(q,p)\} = \{P_i (q,p), P_j(q,p)\} = 0 \)
ii) \( \{Q_i(q,p), P_j(q,p)\} = \delta_{ij} \)