A Trilinear Three-Body Problem

In this paper we present a simplified model of a three-body problem. Place three parallel lines in the plane. Place one mass on each of the lines and let their positions evolve according to Newton's inverse square law of gravitation. We prove the KAM theory applies to our model and simulations are presented. We argue that this model provides an ideal, accessible entry point into the beautiful mathematics involved in the study of the three-body problem.

[1]  John Guckenheimer,et al.  Dstool: Computer assisted exploration of dynamical systems , 1992 .

[2]  Kenneth R. Meyer,et al.  Introduction to Hamiltonian Dynamical Systems and the N-Body Problem , 1991 .

[3]  V. I. Arnolʹd,et al.  Ergodic problems of classical mechanics , 1968 .

[4]  C. Robinson.  Dynamical Systems: Stability, Symbolic Dynamics, and Chaos , 1994 .

[5]  V. M. Tikhomirov,et al.  The General Theory of Dynamical Systems and Classical Mechanics , 1991 .

[6]  P. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[7]  N. G. Parke,et al.  Ordinary Differential Equations. , 1958 .

[8]  S. Smale.  Diffeomorphisms with Many Periodic Points , 1965 .

[9]  J. Kovalevsky,et al.  Lectures in celestial mechanics , 1989 .

[10]  Kenneth R. Meyer,et al.  The Collinear Three-Body Problem with Negative Energy , 1995 .

[11]  Samuel R. Kaplan.  Symbolic Dynamics of the Collinear Three-Body Problem , 1999 .

[12]  V. I. Arnol'd,et al.  PROOF OF A THEOREM OF A.?N.?KOLMOGOROV ON THE INVARIANCE OF QUASI-PERIODIC MOTIONS UNDER SMALL PERTURBATIONS OF THE HAMILTONIAN , 1963 .

[13]  J. Moser.  On invariant curves of area-preserving mappings of an anulus , 1962 .

[14]  J. Zukas.  Introduction to the Modern Theory of Dynamical Systems , 1998 .

[15]  École d'été de physique théorique,et al.  Comportement chaotique des systèmes déterministes : Les Houches, session XXXVI, 29 juin-31 juillet 1981 = Chaotic behaviour of deterministic systems , 1983 .

[16]  Simon Schaffer,et al.  Never at rest. A biography of Isaac Newton , 1981, Medical History.

[17]  J. Hale,et al.  Dynamics and Bifurcations , 1991 .

[18]  Jaume Llibre,et al.  Oscillatory solutions in the planar restricted three-body problem , 1980 .

[19]  Richard McGehee,et al.  Triple collision in the collinear three-body problem , 1974 .

[20]  P. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[21]  L. Perko.  Differential Equations and Dynamical Systems , 1991 .

[22]  J. Barrow-Green.  Poincare and the Three Body Problem , 1996 .