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We consider the problem of three body motion for a relativisticone-dimensional self-gravitating system. After describing the canonicaldecomposition of the action, we find an exact expression for the 3-bodyHamiltonian, implicitly determined in terms of the four coordinate and momentumdegrees of freedom in the system. Non-relativistically these degrees of freedomcan be rewritten in terms of a single particle moving in a two-dimensionalhexagonal well. We find the exact relativistic generalization of thispotential, along with its post-Newtonian approximation. We then specialize tothe equal mass case and numerically solve the equations of motion that followfrom the Hamiltonian. Working in hexagonal-well coordinates, we obtainingorbits in both the hexagonal and 3-body representations of the system, and plotthe Poincare sections as a function of the relativistic energy parameter $\eta$. We find two broad categories of periodic and quasi-periodic motions that werefer to as the annulus and pretzel patterns, as well as a set of chaoticmotions that appear in the region of phase-space between these two types.Despite the high degree of non-linearity in the relativistic system, we findthat the the global structure of its phase space remains qualitatively the sameas its non-relativisitic counterpart for all values of $\eta $ that we couldstudy. However the relativistic system has a weaker symmetry and so itsPoincare section develops an asymmetric distortion that increases withincreasing $\eta $. For the post-Newtonian system we find that it experiences aKAM breakdown for $\eta \simeq 0.26$: above which the near integrable regionsdegenerate into chaos.Comment: latex, 65 pages, 36 figures, high-resolution figures available upon  reques

Chaos in an exact relativistic three-body self-gravitating system