Difference between revisions of "User:Nik"
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\( \{Q_i(q,p), P_j(q,p)\} = \delta_{ij} \) | \( \{Q_i(q,p), P_j(q,p)\} = \delta_{ij} \) | ||
+ | ===Attempts=== | ||
+ | - Use of hamilton's equations from script page 59. I asked Raisa if we are allowed to use things from the script without proof, no answer yet (31.12.14 - 21:00) | ||
+ | The problem there was, that you have to use a hamiltonian from the new coordinate system. May work, haven't checked that exactly. | ||
+ | |||
+ | - Use of simply the chain rule and insert all the things. Some solutions i saw with that said that \( \frac{\partial \Phi}{\partial p} = 0 \) but since \(\Phi \) depends on \(Q \) whicht depends on \(p \) I am not convinced with that. | ||
+ | |||
+ | - Use of simply the chain rule as before. At some point of this solution you should show that \( \frac{\partial Q_a}{\partial q_j}\frac{\partial}{\partial p_j}\frac{\partial}{\partial Q_j}\Phi (q, Q) = 0 \) which nobody has until now. | ||
=== Solution for i) === | === Solution for i) === | ||
=== Solution for ii) === | === Solution for ii) === |
Revision as of 19:57, 31 December 2014
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) \( \{Q_i(q,p), P_j(q,p)\} = \delta_{ij} \)
Attempts
- Use of hamilton's equations from script page 59. I asked Raisa if we are allowed to use things from the script without proof, no answer yet (31.12.14 - 21:00) The problem there was, that you have to use a hamiltonian from the new coordinate system. May work, haven't checked that exactly.
- Use of simply the chain rule and insert all the things. Some solutions i saw with that said that \( \frac{\partial \Phi}{\partial p} = 0 \) but since \(\Phi \) depends on \(Q \) whicht depends on \(p \) I am not convinced with that.
- Use of simply the chain rule as before. At some point of this solution you should show that \( \frac{\partial Q_a}{\partial q_j}\frac{\partial}{\partial p_j}\frac{\partial}{\partial Q_j}\Phi (q, Q) = 0 \) which nobody has until now.