Chaotic particle motion around a homogeneous circular ring

Abstract

We consider test particle motion in a gravitational field generated by a homogeneous circular ring placed in $n$-dimensional Euclidean space. We observe that there exist no stable stationary orbits in $n=6,7,\dots ,10$ but exist in $n=3,4,5$ and clarify the regions in which they appear. In $n=3$, we show that the separation of variables of the Hamilton-Jacobi equation does not occur though we find no signs of chaos for stable bound orbits. Since the system is integrable in $n=4$, no chaos appears. In $n=5$, we find some chaotic stable bound orbits. Therefore, this system is nonintegrable at least in $n=5$ and suggests that the timelike geodesic system in the corresponding black ring spacetimes is nonintegrable.

21 pages, 4 figures; v2: minor revision, add references; v3: published version

References in corpus (12)

• Free motion around black holes with discs or rings: between integrability and chaos - II
• Geodesic Motion in the Singly Spinning Black Ring Spacetime
• Carter-like constants of the motion in Newtonian gravity and electrodynamics
• Constants of Motion for Constrained Hamiltonian Systems: A Particle around a Charged Rotating Black Hole
• Stable Bound Orbits around Black Rings
• Chaos in Geodesic Motion around a Black Ring
• Particle Motion in the Rotating Black Ring Metric
• Free motion around black holes with discs or rings: between integrability and chaos -- VI. The Melnikov method
• A Newtonian analogue of Kerr black hole
• On geometry of deformed black holes: II. Schwarzschild hole surrounded by a Bach-Weyl ring
• Integrability of Particle System around a Ring Source as the Newtonian Limit of a Black Ring
• Particle dynamics in the Newtonian potential sourced by a homogeneous circular ring

Cited by in corpus (1)

• Stable circular orbits in higher-dimensional multi-black hole spacetimes