  We show that if K: P \to R is an autonomous Hamiltonian on a symplectic
manifold (P,\Omega) which attains 0 as a Morse-Bott nondegenerate minimum along
a symplectic submanifold M, and if c_1(TP)|_M vanishes in real cohomology, then
the Hamiltonian flow of K has contractible periodic orbits with bounded period
on all sufficiently small energy levels. As a special case, if the geodesic
flow on the cotangent bundle of M is twisted by a symplectic magnetic field
form, then the resulting flow has contractible periodic orbits on all low
energy levels. These results were proven by Ginzburg and G\"urel when \Omega|_M
is spherically rational, and our proof builds on their work; the argument
involves constructing and carefully analyzing at the chain level a version of
filtered Floer homology in the symplectic normal disc bundle to M.
