I came across the following problem:
Let $(T^{2n}$, $\omega=\sum_{n=0}^\infty dx^k\wedge dy^k)$ be the $2n$ torus and $L\subset T^{2n}$, $L:= \{(x,y)\in T^{2n}|y=0\}$ the lagrangian torus.
a) Show that $\frac{\partial}{\partial x_1}$ is a symplectic vector field on $T^{2n}$.
b) Find a symplectic vector field $V$ on $T^{2n}$, whose flow $\psi_t$ has the property $\psi_1(L)\cap L=\emptyset$.
c) Let $H_t$ be a hamiltonian function on $T^{2n}$ with $\varphi_t$ the associated flow. Show that $\varphi_1(L)\cap L\neq\emptyset$.
Part a) should be easily dealt with, if I'm not mistaken it's pretty obvious that $i_{\frac{\partial}{\partial x_1}}\omega$ is closed.
Then I'm pretty lost. It's obvious from the way the problem is presented that I need a symplectic but not hamiltonian vector field. Further (which is easily seen in the normal torus) the vector field described in a) is not hamiltonian, which would suggest that this is the exact vector field I'm looking for in b). The issue I'm having at the moment is actually proving that to be the case.
c) on the other hand I have absolutely no idea how I would go about actually proving the statement.
Any help would be appreciated.