Suppose $M$ is a symplectic manifold and $\omega_0$ and $\omega_1$ are two symplectic forms belonging to the same de Rham cohomology class. Assume in addition they can be connected via a path of symplectic forms lying in this same class.
Suppose $X$ is a vector field and $f :[0,1] \to \mathbb{R}_+$ is a positive smooth function. Finally, suppose that $f(0) X$ is a Hamiltonian vector field for $\omega_0$ and $f (1)X$ is a Hamiltonian vector field for $\omega_1$.
Question: When is it possible to find a path $\omega_t$ of symplectic forms connecting $\omega_0$ and $\omega_1$ such that $f(t)X$ is a Hamiltonian vector field for $\omega_t$ for each $t \in [0,1]$?
