We got as part of homework for Mechanics this exercise. As the course was a little chaotic I barely got grip of some notions and I feel a bit lost so any solution would be welcomed.
Let V be a finite dimensional manifold with cotangent bundle T*(V): By α denote the canonical 1-form on T*(V), while ω := dα. Given a diffeomorphism Φ : T*(V) -> T*(V); show that the following two statements are equivalent:
- Φ is canonical, i.e.Φ*ω = ω
- Locally there exists a function S : T*(V) ->\mathbb{R}; such that dS = Φ*α - α
$\Phi^*\omega=\omega$ is equivalent to $\Phi^*d\alpha=d\Phi^*\alpha=d\alpha$, this is equivalent to $d(\Phi^*\alpha-\alpha)=0$. Let $U$ be a contractible open neighboorhood diffeomorphic to a ball, the Poincare lemma implies that the fact that the restriction of $\Phi^*\alpha-\alpha$ on a contractible open subset $U$ is closed is equivalent to the existence of a function $S$ defined on $U$ such that $dS=\Phi^*\alpha-\alpha$.
Let $U$ be an open subset diffeomorphic to an open ball, suppose that there exists a function $S$ defined on $U$ such that $dS=\Phi^*\alpha-\alpha$, $0=d(dS)=d(\Phi^*\alpha-\alpha)$.
This implies that $d(\Phi^*\alpha-\alpha)=0$ and $d(\Phi^*\alpha)=\Phi^*d\alpha=d\alpha$, since $\omega=d\alpha$, we deduce that $\Phi^*\omega=\omega$.