Integrable Hamiltonian systems with a periodic orbit or invariant torus unique in the whole phase space

TL;DR

This paper constructs examples of integrable Hamiltonian systems that have a unique periodic orbit or invariant torus in the entire phase space, contrasting the typical organization into families.

Contribution

It provides explicit examples of integrable Hamiltonian systems with a single periodic orbit or invariant torus, including non-compact and compact phase space cases, and reversible analogues.

Findings

Existence of integrable systems with a unique periodic orbit.

Existence of integrable systems with a unique invariant torus.

Construction of reversible analogues of these systems.

Abstract

It is very well known that periodic orbits of autonomous Hamiltonian systems are generically organized into smooth one-parameter families (the parameter being just the energy value). We present a simple example of an integrable Hamiltonian system (with an arbitrary number of degrees of freedom greater than one) with a unique periodic orbit in the phase space (which is not compact). Similar examples are given for Hamiltonian systems with a unique invariant torus (of any prescribed dimension) carrying conditionally periodic motions. Parallel examples for Hamiltonian systems with a compact phase space and with uniqueness replaced by isolatedness are also constructed. Finally, reversible analogues of all the examples are described.