Astronomy and Astrophysics – Astrophysics – General Relativity and Quantum Cosmology
Scientific paper
2003-01-23
Phys.Rev. E69 (2004) 016214
Astronomy and Astrophysics
Astrophysics
General Relativity and Quantum Cosmology
latex, 65 pages, 36 figures, high-resolution figures available upon request
Scientific paper
10.1103/PhysRevE.69.016214
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.
Burnell F. J.
Malecki Jacek
Mann Robert B.
Ohta Taisuke
No associations
LandOfFree
Chaos in an Exact Relativistic 3-body Self-Gravitating System does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Chaos in an Exact Relativistic 3-body Self-Gravitating System, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Chaos in an Exact Relativistic 3-body Self-Gravitating System will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-709551