are required to obtain chaotic behavior. One can draw parallels between the motion of a passive tracer particle in the velocity field of three vortices and the restricted three-body problem of Newtonian mechanics. The gravitational three-body problem has also been studied using general relativity. Physically, a relativistic treatment becomes necessary in systems with very strong gravitational fields, such as near the event horizon of a black hole. However, the relativistic problem is considerably more difficult than in Newtonian mechanics, and sophisticated numerical techniques are required. Even the full two-body problem (i.e. for arbitrary ratio of masses) does not have a rigorous analytic solution in general relativity. -body problem The three-body problem is a special case of the -body problem, which describes how objects move under one of the physical forces, such as gravity. These problems have a global analytical solution in the form of a convergent power series, as was proven by Karl F. Sundman for and by Qiudong Wang for (see -body problem for details). However, the Sundman and Wang series converge so slowly that they are useless for practical purposes; therefore, it is currently necessary to approximate solutions by numerical analysis in the form of numerical integration or, for some cases, classical trigonometric series approximations (see -body simulation). Atomic systems, e.g. atoms, ions, and molecules, can be treated in terms of the quantum -body problem. Among classical physical systems, the -body problem usually refers to a galaxy or to a cluster of galaxies; planetary systems, such as stars, planets, and their satellites, can also be treated as -body systems. Some applications are conveniently treated by perturbation theory, in which the system is considered as a two-body problem plus additional forces causing deviations from a hypothetical unperturbed two-body trajectory. In popular culture In the classic 1951 science-fiction film The Day the Earth Stood Still, the alien Klaatu, using the pseudonym Mr. Carpenter, makes some annotations to equations on Prof. Barnhardt's blackboard. Those equations are an accurate description of a particular form of the three-body problem. The first volume of Chinese author Liu Cixin's Remembrance of Earth's Past trilogy is titled The Three-Body Problem and features the three-body problem as a central plot device. See also Few-body systems Galaxy formation and evolution Gravity assist Lagrange point Low-energy transfer Michael Minovitch -body simulation Symplectic integrator Sitnikov problem Triple star system References Further reading External links Physicists Discover a Whopping 13 New