  A Hamiltonian circle action on a compact symplectic manifold is known to be a
closed geodesic with respect to the Hofer metric on the group of Hamiltonian
diffeomorphisms. If the momentum map attains its minimum or maximum at an
isolated fixed point with isotropy weights not all equal to plus or minus one,
then this closed geodesic can be deformed into a loop of shorter Hofer length.
In this paper we give a lower bound for the possible amount of shortening, and
we give a lower bound for the index ("number of independent shortening
directions"). If the minimum or maximum is attained along a submanifold B, then
we deform the circle action into a loop of shorter Hofer length whenever the
isotropy weights have sufficiently large absolute values and the normal bundle
of B is sufficiently un-twisted.
