Maximum number of independent constants of motion in involution Consider a holonomic system. Let $M$ be the phase space $T^{\ast}Q$ where $Q$ is the configuration space with $\dim(Q)=n$. Two functions $f,g: M \to \mathbb{R}$ are in involution if they Poisson-commute ($\{f,g\}=0$).  I was told that the maximum number of (functionally) independent constanst of motion in involution is $n$, but I can't prove that (or even see why). If you know about this stuff and you're able to prove this fact, thank you so much.
My idea is that it's probably related to the fact that the symplectic form $\omega$ is not-degenerate, but I couldn't get anything from this.

The essence is the following simple fact.
Suppose we have a symplectic form $\omega $ on a $2n$-dim vector space $V$. By Darboux's theorem, we can find a basis of $V$, $\{x_i,y_i | 1\leq i \leq n \}$, such that
$$\omega(x_i,y_j) = \delta_{ij}, \quad \omega(x_i,x_j) =\omega(y_i,y_j )=0. $$
We have actually a direct decomposition of $V$, $V= \oplus_{i=1}^n V_i $, with $V_i$ the 2-dim subspace spanned by $(x_i,y_i )$.
Now consider a subspace $W$ which is of at least $n+1$ dim. We have
$$W = \oplus_{i=1}^n ( W \cap V_i ). $$
Counting the dimension, we know that at least for some $i$, the intersection $W\cap V_i $ is of dim 2. That is, $V_i \subset W$. By $\omega $ is nonzero on $V_i $.
