  This paper studies the question of when a loop $\phi$ in the group
Symp$(M,\omega)$ of symplectomorphisms of a symplectic manifold $(M,\omega)$ is
isotopic to a loop that is generated by a time-dependent Hamiltonian function.
(Loops with this property are said to be Hamiltonian.) Our main result is that
Hamiltonian loops are rigid in the following sense: if $\phi$ is Hamiltonian
with respect to $\omega$, and if $\phi'$ is a small perturbation of $\phi$ that
preserves another symplectic form $\omega'$, then $\phi'$ is Hamiltonian with
respect to $\omega'$. This allows us to get some new information on the
structure of the flux group, i.e. the image of $\pi_1(Symp(M,\omega))$ under
the flux homomorphism. We give a complete proof of our result for some
manifolds, and sketch the proof in general. The argument uses methods developed
by Seidel for studying properties of Hamiltonian loops via the quantum homology
of $M$.
