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In physics and classical mechanics, the three-body problem is the problem of taking an initial set of data that specifies the positions, masses and velocities of three bodies for some particular point in time and then determining the motions of the three bodies, in accordance with the laws of classical mechanics (Newton's laws of motion and of universal gravitation). The three-body problem is a special case of the ''n''-body problem. Historically, the first specific three-body problem to receive extended study was the one involving the Moon, the Earth and the Sun.〔(【引用サイトリンク】title=Historical Notes: Three-Body Problem )〕 In an extended modern sense, a three-body problem is a class of problems in classical or quantum mechanics that model the motion of three particles. ==History== The gravitational problem of three bodies in its traditional sense dates in substance from 1687, when Isaac Newton published his "Principia" (''Philosophiæ Naturalis Principia Mathematica''). In Proposition 66 of Book 1 of the "Principia", and its 22 Corollaries, Newton took the first steps in the definition and study of the problem of the movements of three massive bodies subject to their mutually perturbing gravitational attractions. In Propositions 25 to 35 of Book 3, Newton also took the first steps in applying his results of Proposition 66 to the lunar theory, the motion of the Moon under the gravitational influence of the Earth and the Sun. The problem became of technical importance in the 1720s, as an accurate solution would be applicable to navigation, specifically for the determination of longitude at sea. This problem was addressed by Amerigo Vespucci and by Galileo Galilei before being solved by John Harrison's invention of the Marine chronometer. Before the chronometer became available, Vespucci had used, in 1499, knowledge of the position of the moon to determine his position in Brazil. However the accuracy of the lunar theory was low, due to the perturbing effect of the Sun, and planets, on the motion of the Moon around the Earth. Jean d'Alembert and Alexis Clairaut, who developed a longstanding rivalry, both attempted to analyze the problem in some degree of generality, and by the use of differential equations to be solved by successive approximations. They submitted their competing first analyses to the Académie Royale des Sciences in 1747.〔The 1747 memoirs of both parties can be read in the volume of ''Histoires'' (including ''Mémoires'') of the Académie Royale des Sciences for 1745 (belatedly published in Paris in 1749) (in French): ::Clairaut: "On the System of the World, according to the principles of Universal Gravitation" (at pp. 329–364); and ::d'Alembert: "General method for determining the orbits and the movements of all the planets, taking into account their mutual actions" (at pp. 365–390). :The peculiar dating is explained by a note printed on page 390 of the 'Memoirs' section:"Even though the preceding memoirs, of Messrs. Clairaut and d'Alembert, were only read during the course of 1747, it was judged appropriate to publish them in the volume for this year" (i.e. the volume otherwise dedicated to the proceedings of 1745, but published in 1749).〕 It was in connection with these researches, in Paris, in the 1740s, that the name "three-body problem" (Problème des Trois Corps) began to be commonly used. An account published in 1761 by Jean d'Alembert indicates that the name was first used in 1747.〔Jean d'Alembert, in a paper of 1761 reviewing the mathematical history of the problem, mentions that Euler had given a method for integrating a certain differential equation "in 1740 (seven years before there was question of the Problem of Three Bodies)": see d'Alembert, "Opuscules Mathématiques", vol.2, Paris 1761, Quatorzième Mémoire ("Réflexions sur le Problème des trois Corps, avec de Nouvelles Tables de la Lune ...") pp. 329–312, at sec. VI, p. 245.〕 In 1887, mathematicians Heinrich Bruns and Henri Poincaré showed that there is no general analytical solution for the three-body problem given by algebraic expressions and integrals. The motion of three bodies is generally non-repeating, except in special cases. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Three-body problem」の詳細全文を読む スポンサード リンク
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