Restricted Three-Body Problem -- from Eric Weisstein's World of Physics

The three-body problem considers three mutually interacting masses , , and . In the restricted three-body problem, is taken to be small enough so that it does not influence the motion of and , which are assumed to be in circular orbits about their center of mass. The orbits of three masses are further assumed to all lie in a common plane. If and are in elliptical instead of circular orbits, the problem is variously known as the "elliptic restricted problem" or "pseudorestricted problem" (Szebehely 1967, pp. 30 and 39).

The efforts of many famous mathematicians have been devoted to this difficult problem, including Euler Eric Weisstein's World of Biography and Lagrange Eric Weisstein's World of Biography (1772), Jacobi Eric Weisstein's World of Biography (1836), Hill (1878), Poincaré Eric Weisstein's World of Biography (1899), Levi-Civita (1905), and Birkhoff (1915). In 1772, Euler first introduced a synodic (rotating) coordinate system. Jacobi (1836) subsequently discovered an integral of motion in this coordinate system (which he independently discovered) that is now known as the Jacobi integral. Hill (1878) used this integral to show that the Earth-Moon distance remains bounded from above for all time (assuming his model for the Sun-Earth-Moon system is valid), and Brown (1896) gave the most precise lunar theory of his time.

Poincaré published his monumental Méthodes Nouvelles in 1899, emphasizing qualitative aspects of celestial mechanics, including modern concepts such at phase space surfaces of section. Birkhoff (1915) further developed these qualitative methods. The important problem of regularization was considered by Thiele (1892), Painlevé (1897), Levi-Civita (1903), Burrau (1906), Sundman (1912), and Birkhoff (1915). Painlevé Eric Weisstein's World of Biography proved that all singularities are collisions for n = 3. Sundman found a uniformly convergent infinite series involving known functions that "solves" the restricted three-body problem in the whole plane (once singularities are removed through the process of regularization). Since such global regularizations are available for this problem, the restricted problem of three bodies can be considered to be complete "solved." However, this "solution" does not address issues of stability, allowed regions of motion, and so on, and so is of limited practical utility (Szebehely 1967, p. 42). Furthermore, an unreasonably large number of terms (of order ) of Sundman's series are required in to attain anything like the accuracy required for astronomical observations.

Copenhagen Problem, Jacobi Integral, Lagrange Points, n-Body Problem, Three-Body Problem, Two-Body Problem

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