Construction of a non-autonomous Hamiltonian diffeomorphism

Question

Let $(M,\omega)$ be a symplectic manifold. I have read that the autonomous Hamiltonian diffeomorphisms (i.e. a Hamiltonian diffeomorphism generated by a time-independant Hamiltonian) form a proper subset of the Hamiltonian diffeomorphisms of $M$, but I haven't been able to find a proof of that. Does anyone have a link in which a non-autonomous Hamiltonian diffeomorphism is constructed, and proven that it is indeed non-autonomous.

Answer 1

Here's a hint: use the fact that the exponential map for $\mathrm{Sp}(2n)$ is not surjective, and the fact that if $\phi$ is a Hamiltonian diffeomorphism coming from a time-independent Hamiltonian, then at a fixed point $p$ of $\phi$, the differential $D_p \phi \in \mathrm{Sp}(T_p M)$ is in the image of the exponential map.

If you'd like a reference where some counterexamples are stated in the form of exercises, take a look at McDuff-Salamon, Introduction to Symplectic Topology, Exercise 3.1.16 on p. 103 of 3rd Ed.