Periodic orbits in the Kepler-Heisenberg problem

Abstract

The Kepler-Heisenberg problem is that of determining the motion of a planet around a sun in the Heisenberg group, thought of as a three-dimensional sub-Riemannian manifold. The sub-Riemannian Hamiltonian provides the kinetic energy, and the gravitational potential is given by the fundamental solution to the sub-Laplacian. The dynamics are at least partially integrable, possessing two first integrals as well as a dilational momentum which is conserved by orbits with zero energy. The system is known to admit closed orbits of any rational rotation number, which all lie within the fundamental zero-energy integrable subsystem. Here we demonstrate that, under mild conditions, zero-energy orbits are self-similar. Consequently we find that these zero-energy orbits stratify into three families: future collision, past collision, and quasi-periodicity, with all collisions occurring in finite time.

An introduction to the Kepler-Heisenberg problem

“…Periodic orbits. The existence of periodic orbits was established in [11]. However, the proof followed the direct method in the calculus of variations and provided very little information about such orbits beyond their existence.…”

Non-integrability of the Kepler and the two body problem on the Heisenberg group

“…where κ is a non-zero real parameter. The properties of this system have been analysed in [9,12], where it was shown that is has an invariant submanifold given by z = 0, p z = 0, p θ = xp y − yp x = 0 on which all trajectories are straight lines. More importantly, it was also demonstrated that all periodic solutions must lie on the zeroenergy level and that the whole system is Liouville integrable there.…”

Non-Integrability of the Kepler and the Two-Body Problems on the Heisenberg Group

“…where κ is a non-zero real parameter. The properties of this system have been analysed in [4,9,12], where it was shown that it has an invariant submanifold given by z = 0, p z = 0, p θ = xp y − yp x = 0 on which all trajectories are straight lines. More importantly, it was also demonstrated that all periodic solutions must lie on the zero-energy level and that the whole system is Liouville integrable there.…”