Hamiltonian flow local diffeomorphism?

Question

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 understandingSo 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.

Answer 1

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.