I have the following question:
Let $(M,\omega)$ be a symplectic manifold and let $N_1$ and $N_2$ be submanifolds of $M$ such that there is a diffeomorphism $\psi:M \rightarrow M$ such that $\psi(N_1) = N_2$. Does there exist a symplectomorphism $\varphi:M \rightarrow M$ such that $\varphi(N_1) = N_2$?
Can anyone give me a hint? I'm really struggling. Thanks!
As the simplest example I can think of, take a loop in $S^2$. There's a diffeomorphism taking any two loops to another; but the area of the two sides of the loop are preserved by a symplectomorphism (because symplectomorphisms are in particular volume-preserving).
More in line with my comment above, consider $T^*M$ with the standard form. The zero section is a Lagrangian submanifold, and you can easily check that a section (considered as a 1-form $\omega$) is Lagrangian if and only if $\omega$ is closed. But all sections are smoothly isotopic, whence there is a diffeomorphism taking one to another.