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In physics, the ''n''-body problem is the problem of predicting the individual motions of a group of celestial objects interacting with each other gravitationally.〔Leimanis and Minorsky: Our interest is with Leimanis, who first discusses some history about the ''n''-body problem, especially Ms. Kovalevskaya's ~1868-1888, twenty-year complex-variables approach, failure; Section 1: The Dynamics of Rigid Bodies and Mathematical Exterior Ballistics (Chapter 1, ''the motion of a rigid body about a fixed point'' (Euler and Poisson ''equations''); Chapter 2, ''Mathematical Exterior Ballistics''), good precursor background to the ''n''-body problem; Section 2: Celestial Mechanics (Chapter 1, ''The Uniformization of the Three-body Problem'' (Restricted Three-body Problem); Chapter 2, ''Capture in the Three-Body Problem''; Chapter 3, ''Generalized n-body Problem'').〕 Solving this problem has been motivated by the desire to understand the motions of the Sun, Moon, planets and the visible stars. In the 20th century, understanding the dynamics of globular cluster star systems became an important ''n''-body problem.〔See references cited for Heggie and Hut.〕 The ''n''-body problem in general relativity is considerably more difficult to solve. The classical physical problem can be informally stated as: ''given the quasi-steady orbital properties'' (''instantaneous position, velocity and time'')〔''Quasi-steady'' loads refers to the instantaneous inertial loads generated by instantaneous angular velocities and accelerations, as well as translational accelerations (9 variables). It is as though one took a photograph, which also recorded the instantaneous position and properties of motion. In contrast, a ''steady-state'' condition refers to a system's state being invariant to time; otherwise, the first derivatives and all higher derivatives are zero.〕 ''of a group of celestial bodies, predict their interactive forces; and consequently, predict their true orbital motions for all future times''.〔R. M. Rosenberg states the ''n''-body problem similarly (see References): ''Each particle in a system of a finite number of particles is subjected to a Newtonian gravitational attraction from all the other particles, and to no other forces. If the initial state of the system is given, how will the particles move?'' Rosenberg failed to realize, like everyone else, that it is necessary to determine the forces ''first'' before the motions can be determined.〕 To this purpose the two-body problem has been completely solved and is discussed below; as is the famous ''restricted 3-Body Problem''.〔A general, classical solution in terms of first integrals is known to be impossible. An exact theoretical solution for arbitrary ''n'' can be approximated via Taylor series, but in practice such an infinite series must be truncated, giving at best only an approximate solution; and an approach now obsolete. In addition, the ''n''-body problem may be solved using numerical integration, but these, too, are approximate solutions; and again obsolete. See Sverre J. Aarseth's book Gravitational N-body Simulations listed in the References.〕 == History == Knowing three orbital positions of a planet's orbit – positions obtained by Sir Isaac Newton (1643-1727) from astronomer John Flamsteed〔See David H. and Stephen P. H. Clark's The Suppressed Scientific Discoveries of Stephen Gray and John Flamsteed, Newton's Tyranny, ''W. H. Freeman and Co''., 2001. A popularization of the historical events and bickering between those parties, but more importantly about the results they produced.〕 – Newton was able to produce an equation by straightforward analytical geometry, to predict a planet's motion; i.e., to give its orbital properties: position, orbital diameter, period and orbital velocity.〔See "''Discovery of gravitation'', A.D. 1666" by Sir David Brewster, in The Great Events by Famous Historians, Rossiter Johnson, LL.D. Editor-in-Chief, Volume XII, pp. 51-65, ''The National Alumni'', 1905.〕 Having done so he and others soon discovered over the course of a few years, those equations of motion did not predict some orbits very well or even correctly.〔Rudolf Kurth has an extensive discussion in his book (see References) on planetary perturbations. An aside: these mathematically undefined planetary perturbations (wobbles) still exist undefined even today and planetary orbits have to be constantly updated, usually yearly. See Astronomical Ephemeris and the American Ephemeris and Nautical Almanac, prepared jointly by the Nautical Almanac Offices of the United Kingdom and the United States of America.〕 Newton realized it was because gravitational interactive forces amongst all the planets was affecting all their orbits. The above discovery goes right to the heart of the matter as to what exactly the ''n''-body problem is physically: as Newton realized, it is not sufficient to just specify the initial position and velocity, or three orbital positions either, to determine a planet's true orbit: ''the gravitational interactive forces have to be known too''. Thus came the awareness and rise of the n-body “problem” in the early 17th century. These gravitational attractive forces do conform to Newton's ''Laws of Motion'' and to his ''Law of Universal Gravitation'', but the many multiple (''n''-body) interactions have historically made any exact solution intractable. Ironically, this conformity led to the wrong approach. After Newton's time the ''n''-body problem historically was not stated correctly ''because it did not include a reference to those gravitational interactive forces''. Newton does not say it directly but implies in his Principia the ''n''-body problem is unsolvable because of those gravitational interactive forces.〔See Principia, Book Three, ''System of the World'', "''General Scholium''," page 372, last paragraph. Newton was well aware his math model did not reflect physical reality. This edition referenced is from the Great Books of the Western World, Volume 34, which was translated by Andrew Motte and revised by Florian Cajori. This same paragraph is on page 1160 in Stephen Hawkins' huge On the Shoulders of Giants, 2002 edition; is a copy from Daniel Adee's 1848 addition. Cohen also has translated new editions: Introduction to Newton's 'Principia' , 1970; and Isaac Newton's ''Principia'', with Varian Readings, 1972. Cajori also wrote a History of Science, which is on the Internet.〕 Newton said〔 in his Principia, paragraph 21: Newton concluded via his 3rd Law that "according to this Law all bodies must attract each other." This last statement, which implies the existence of gravitational interactive forces, is key. As shown below, the problem also conforms to Jean Le Rond D'Alembert's non-Newtonian 1st and 2nd ''Principles'' and to the nonlinear ''n''-body problem algorithm, the latter allowing for a closed form solution for calculating those interactive forces. The problem of finding the general solution of the ''n''-body problem was considered very important and challenging. Indeed in the late 19th century King Oscar II of Sweden, advised by Gösta Mittag-Leffler, established a prize for anyone who could find the solution to the problem. The announcement was quite specific: In case the problem could not be solved, any other important contribution to classical mechanics would then be considered to be prize-worthy. The prize was awarded to Poincaré, even though he did not solve the original problem. (The first version of his contribution even contained a serious error〔for details of the serious error in Poincare's first submission see the article by Diacu〕). The version finally printed contained many important ideas which led to the development of chaos theory. The problem as stated originally was finally solved by Karl Fritiof Sundman for ''n'' = 3. 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「N-body problem」の詳細全文を読む スポンサード リンク
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