Straight line orbits in Hamiltonian flows

We investigate straight-line orbits (SLO) in Hamiltonian force fields using both direct and inverse methods. A general theorem is proven for natural Hamiltonians quadratic in the momenta for arbitrary dimensions and is considered in more detail for two and three dimensions. Next we specialize to homogeneous potentials and their superpositions, including the familiar Hénon–Heiles problem. It is shown that SLO’s can exist for arbitrary finite superpositions of N-forms. The results are applied to a family of potentials having discrete rotational symmetry as well as superpositions of these potentials.