On the intersections of projected Hamiltonian orbits in cotangent bundles

We study the generic behaviour of Hamiltonian trajectories on a regular level set in the cotangent bundle, after projection to the base. We prove that for a generic submersive level set, projected trajectories have discrete (self-)intersections. Additionally, fixing end-point fibres, we prove that all intersections can be perturbed away if the base has dimension 3 ⩾ 3 . In particular, this applies to periodic orbits, and both results hold for Reeb flows on fibre-wise star-shaped hypersurfaces, including non-reversible Finsler flows, which answers a question of Rademacher. In the proof we make use of a multi-jet transversality theorem.