We have $H(q,Q,t)$. There is unique solution $Q$ for $p_i = \frac{\partial H}{\partial q_i}$ and we have $P_i = - \frac{\partial H}{\partial Q_i}$.
We want to prove from only this that the fundamental Poisson Bracket equations are fulfilled.
These are: $$ \{Q_i, P_j\} = \delta_{i,j}$$ $$ \{Q_i, Q_j\} = 0$$ $$ \{P_i, P_j\} = 0$$
where $$\{f, g\} = \sum_{k=1}^{n} \left(\frac{\partial f}{\partial q_k}\frac{\partial g}{\partial p_k} - \frac{\partial f}{\partial p_k}\frac{\partial g}{\partial q_k}\right)$$
The solution:
I think I have proven $ \{P_i, P_j\} = 0$ by using the definition for $P$, then using the unique solution and then applying the multi-dimensional chain rule (here the sum is eliminated). As in all of this, as the system is continuously differentiable, we exchange partial derivatives, I get $$ ...= \frac{\partial H}{\partial Q_i \partial Q_j} - \frac{\partial H}{{\partial Q_j \partial Q_i}} = 0$$
I'm having trouble with $$ \{Q_i, Q_j\} = \sum_{k=1}^{n} \left(\frac{\partial Q_i}{\partial q_k}\frac{\partial Q_j}{\partial p_k} - \frac{\partial Q_i}{\partial p_k}\frac{\partial Q_j}{\partial q_k} \right)= ?$$
There doesn't seem to be any relation I can use.
My question: What underlying principal or relationship exists here that I can use to proof this and the Dirac delta identity?
2nd question: May I use the chain rule in the first proof and is this way correct or how is it usually done?
