I am currently reading Arnold's proof of the Darboux theorem in his book on classical mechanics and fail to understand some point.


**The background**
So he wants to show that any symplectic form is locally (in a neighbourhood of some $x$) $$\omega = \sum\_{i=1}^{n} dp\_i \wedge dq\_i.$$
It's on p. 230 in his book on classical mechanics.
Now, we says that the first coordinate can be chosen as a non-constant linear function $p\_1$. Now, we pick the one that satisfies $p\_1(x)=0.$


Since there is a canonical isomorphism between tangent and cotangent space in symplectic geometry, we can get a vector field $P\_1 = I dp\_1.$ Now since $dp\_1$ is non-trivial $P\_1(x) \neq 0.$ This is a proper vector in the tangent space at the point $x$ and hence we can consider the orthogonal complement $N^{2n-1}$ to it.


Now, I am getting a little bit confused: He says, consider the hamiltonian flow $P\_1^t$ with Hamilton function $p\_1.$ We consider the time necessary to go from $N$ (I guess this is $N^{2n-1}$) to the point $z = P\_1^t(y)$ for $y \in N.$ Under the action of $P\_1^t$ as a function of the point $z$. By the usual theorems in the theory of ODEs, this function is defined and differentiable in a neighbourhood of the point $x \in \mathbb{R}^{2n}.$ Denote it by $q\_1.$


**My understanding**So let me summarize: He says that there is a function $q\_1$ that measures the time it takes to go from the hypersurface $N^{2n-1}$ to any point $z$ in the local neighbourhood and he says that this function is properly defined locally? 


**The question** Could anybody explain why we can reach any point locally and i.e. why we cannot reach a point on several ways at different times (if we can, maybe we take the shortest time, but anyway.) I would really love to see a motivation for this construction or some additional remarks. 



Hamiltonian flow local diffeomorphism?
I am not sure that it is a research level question but since when I was a PhD student I also asked me this question I give an answer: 


we work in a very small neighborhood and consider the following mapping 
$\phi: N^{2n-1}\times R \to \mathbb{R}^{2n}$ sending the point $(y, t)$ to the point $P^1\_t(y)$. This is a smooth mapping, its differential at the point $(x,0)$ is nondegenerate (to do it, just construct the matrix of the differential at the point $(x,0)$); then it is a local diffeomorphism which implies it is locally bijective and injective as you want us to explain. 


The additional remark is that I do not know any source where the proof of the Darboux theorem is explained nicely and self-contained and also explains the geometry behind the proof. 


