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

Floer homology in disk bundles and symplectically twisted geodesic flows