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.
A fundamental fact about smooth vector fields is that if $x$ is any point where $P_1(x)\ne 0$, there exist smooth coordinates $(u^1,\dots,u^n)$ defined on a neighborhood $U$ of $x$ in which $P_1$ has the coordinate representation $\partial/\partial u^1$, and hence its flow is given by $$P_1^t(t,(u^1,\dots,u^n)) = (u^1+t,u^2,\dots,u^n).$$ (See, for example, my Introduction to Smooth Manifolds (2nd ed.), Theorem 9.22.)
Arnold is a little unclear about the exact definition of his hypersurface $N$, but evidently what he means to say is that we can take $N$ to be the smooth hypersurface in $M$ given by the "orthogonal complement" to $P_1(x)$ in coordinates, not a hyperplane in the tangent space. (He also remarks that we could have taken $N$ to be any surface transverse to $P_1(x)$.)
In our coordinates $(u^1,\dots,u^n)$, therefore, we can take $N$ to be the hypersurface defined by $u^1=0$. Shrinking $U$ if necessary, we can also assume that $U$ is a coordinate cube of the form $\{u: |u|<c\}$ for some $c>0$. Then we can reach any point $u=(u^1,\dots,u^n)\in U$ by flowing from the point $(0,u^2,\dots,u^n)\in N$ for time $u^1$, and this is the only integral curve starting on $N$ that stays in $U$ and reaches $u$. The "time necessary to go from $N$ to an arbitrary point in $U$" is just the coordinate function $u^1$, which is well defined and smooth in $U$.
(BTW, a small matter of notation: It's very common in differential geometry to introduce a manifold of dimension $k$ by saying "Let $M^k$ be a manifold." This is to be read as a shorthand for "Let $M$ be a manifold of dimension $k$." The superscript $k$ is not part of the name of the manifold, and is typically omitted after the first mention. This is why Arnold switched from $N^{2n-1}$ to $N$.)