Floer homology in disk bundles and symplectically twisted geodesic flows

We show that if $K: P\to\mathbb{R}$ is an autonomous Hamiltonian on a symplectic manifold $(P,\Omega)$ which attains a Morse-Bott nondegenerate minimum of 0 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 T*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 Gürel 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 disk bundle to $M$.