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.