Chaos in an Exact Relativistic 3-body Self-Gravitating System

Abstract

We consider the problem of three body motion for a relativistic one-dimensional self-gravitating system. After describing the canonical decomposition of the action, we find an exact expression for the 3-body Hamiltonian, implicitly determined in terms of the four coordinate and momentum degrees of freedom in the system. Non-relativistically these degrees of freedom can be rewritten in terms of a single particle moving in a two-dimensional hexagonal well. We find the exact relativistic generalization of this potential, along with its post-Newtonian approximation. We then specialize to the equal mass case and numerically solve the equations of motion that follow from the Hamiltonian. Working in hexagonal-well coordinates, we obtaining orbits in both the hexagonal and 3-body representations of the system, and plot the Poincare sections as a function of the relativistic energy parameter $\eta$. We find two broad categories of periodic and quasi-periodic motions that we refer to as the annulus and pretzel patterns, as well as a set of chaotic motions that appear in the region of phase-space between these two types. Despite the high degree of non-linearity in the relativistic system, we find that the the global structure of its phase space remains qualitatively the same as its non-relativisitic counterpart for all values of $\eta$ that we could study. However the relativistic system has a weaker symmetry and so its Poincare section develops an asymmetric distortion that increases with increasing $\eta$. For the post-Newtonian system we find that it experiences a KAM breakdown for $\eta \simeq 0.26$: above which the near integrable regions degenerate into chaos.

arXiv commentlatex, 65 pages, 36 figures, high-resolution figures available upon request

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Cited by in corpus (7)

• N-body Gravity and the Schroedinger Equation
• 3-Body Dynamics in a (1+1) Dimensional Relativistic Self-Gravitating System
• Perturbative Quantum Gravity Coupled to Particles in (1+1)-Dimensions
• A semi-relativistic approach to the circular restricted three body problem and a numerical experiment around the 3:2 resonance
• Chaos in a 3-body Self-Gravitating Cosmological Spacetime
• One-Dimensional Relativistic Self-Gravitating Systems
• Dynamics of Coulombic and Gravitational Periodic Systems