Straight Line Orbits in Hamiltonian Flows

TL;DR

This paper studies the existence and properties of straight-line periodic orbits in Hamiltonian systems, providing a general theorem and exploring specific potentials like Hénon-Heiles, revealing new classes of such orbits.

Contribution

It proves a general theorem for natural Hamiltonians quadratic in momenta and demonstrates the existence of straight-line orbits in various superpositions of potentials, including symmetric Hénon-Heiles systems.

Findings

SLOs can exist in arbitrary superpositions of N-forms.

Theorem applies to natural Hamiltonians in any dimension.

SLOs are found in generalized Hénon-Heiles potentials with symmetry.

Abstract

We investigate periodic 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 in arbitrary dimension and specialized to two and three dimension. Next we specialize to homogeneous potentials and their superpositions, including the familiar H\'enon-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 generalized H\'enon-Heiles potentials having discrete rotational symmetry. SLO's are also found for superpositions of these potentials.