At the moment I'm struggling with a problem I found in a script to one of my lectures:
Let $\phi \in C^\infty(\mathbb{R}^{2n})$ have the property that the system $p_i=\frac{\partial}{\partial q_i}\phi(q,\tilde{q})$ has a unique smooth solution $\tilde{q}=\tilde{q}(q,p)$.
Define $\tilde{p}_i(q,p)=-\frac{\partial}{\partial\tilde{q}_i}\phi(q,\tilde{q})|_{\tilde{q}=\tilde{q}(q,p)}$. Furthermore, let $\{\cdot, \cdot\}$ be the Poisson bracket, such that $\{f,g\}=\sum^n_{j=1} \frac{\partial f}{\partial q_j}\frac{\partial g}{\partial p_j} - \frac{\partial f}{\partial p_j}\frac{\partial g}{\partial q_j}$.
Show that: $$\{\tilde{q}_i(q,p), \tilde{q}_j\}=0$$ $$\{\tilde{p}_i(q,p), \tilde{p}_j\}=0$$ $$\{\tilde{q}_i(q,p), \tilde{p}_j\}=\delta_{ij}$$ That is, the map $(q,p) \to (\tilde{q}, \tilde{p})$ is canonical.
I have only been able to deduce equations like $$-\frac{\partial^2 \phi}{\partial q_i \partial q_k}(q,\tilde{q}) = \sum^n_{s=1} \frac{\partial^2 \phi}{\partial q_i \partial \tilde{q}_s}(q,\tilde{q})\frac{\partial\tilde{q}_s}{\partial q_k}$$ or $$\frac{\partial\tilde{p}_i}{\partial q_k} = -\frac{\partial^2 \phi}{\partial \tilde{q}_i \partial q_k}(q,\tilde{q}) - \sum^n_{s=1} \frac{\partial^2 \phi}{\partial \tilde{q}_i \partial \tilde{q}_s}(q,\tilde{q})\frac{\partial\tilde{q}_s}{\partial q_k}$$
I was able to deduce some other equations, but those are similar in form and don't seem to help at all. Since I've been stuck for hours now (I think that I just too desperately want to do something with those equations) I would love if someone could point me to a solution. Thanks in advance!