Suppose that we have the symplectic manifold $(M,\omega)$ where $M$ is a certain annulus, let's say of minimum radius $r_1$ and maximum radius $r_2$, and $\omega=dr\wedge d\theta$. There exists a symplectomorphism $\psi:M\rightarrow M$ that switches the boundaries, given by $\psi(r,\theta)=(r_2-r+r_1,-\theta)$. Now I am wondering if this could be an Hamiltonian diffeomorphism ? Does there exist a Hamiltonian function that could give us this symplectomorphism ?
More generally suppose I have a circle of radius $\tilde r$ with $\tilde r<r_1+\frac{r_2-r_1}{2}$. Could there exist a Hamiltonian symplectomorphism that displaces this circle from itself ?
Any insight is appreciated.