The three-body problem Z.E. Musielak1 and B. Quarles2 1Department of Physics, The University of Texas at Arlington, Arlington, TX 76019, USA 2Space Science and Astrobiology Division 245-3, NASA Ames Research Center, Moffett Field, CA 94035, USA E-mail:
[email protected];
[email protected] Abstract. The three-body problem, which describes three masses interacting through Newtonian gravity without any restrictions imposed on the initial positions and velocities of these masses, has attracted the attention of many scientists for more than 300 years. In this paper, we present a review of the three-body problem in the context of both historical and modern developments. We describe the general and restricted (circular and elliptic) three-body problems, different analytical and numerical methods of finding solutions, methods for performing stability analysis, search for periodic orbits and resonances, and application of the results to some interesting astronomical and space dynamical settings. We also provide a brief presentation of the general and restricted relativistic three-body problems, and discuss their astronomical applications. arXiv:1508.02312v1 [astro-ph.EP] 10 Aug 2015 The three-body problem 2 1. Introduction In the three-body problem, three bodies move in space under their mutual gravitational interactions as described by Newton’s theory of gravity. Solutions of this problem require that future and past motions of the bodies be uniquely determined based solely on their present positions and velocities. In general, the motions of the bodies take place in three dimensions (3D), and there are no restrictions on their masses nor on the initial conditions.