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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Painlevé conjecture ： ウィキペディア英語版 Painlevé conjecture In physics, the Painlevé conjecture is a conjecture about singularities among the solutions to the ''n''-body problem: there are noncollision singularities for ''n'' ≥ 4.〔Florin N. Diacu. Painlevé's Conjecture. The Mathematical Intelligencer. Vol 13, no. 2, 1993〕〔Florin Diacu, Philip Holmes, Celestial Encounters: The Origins of Chaos and Stability. Princeton University Press, 1996〕 The conjecture has been proven for ''n'' ≥ 5 by Jeff Xia.〔Zhihong Xia. The Existence of Noncollision Singularities in Newtonian Systems. Annals of Mathematics. Second Series, Vol. 135, No. 3 (May, 1992), pp. 411–468〕〔Donald G. Saari and Zhihong (Jeff) Xia. Off to Infinity in Finite Time. Notices of the AMS, vol 42, no. 5, 1993, pp. 538–546〕 The 4-particle case remains an open problem.〔Edward Belbruno. Capture Dynamics and Chaotic Motions in Celestial Mechanics. Princeton University Press, 2004 p. 8〕 ==Background and statement== Solutions $(\backslash mathbf,\backslash mathbf)$ of the ''n''-body problem $\backslash dot\backslash mathbf,\; \backslash dot)$ (where M are the masses and U denotes the gravitational potential) are said to have a singularity if there is a sequence of times $t\_n$ converging to a finite $t^$ * where $\backslash nabla\; U(\backslash mathbf(t\_n))\backslash rightarrow\; \backslash infty$. That is, the forces and accelerations become infinite at some finite point in time. A ''collision singularity'' occurs if $\backslash mathbf(t)$ tends to a definite limit when $t\backslash rightarrow\; t^$ *, t*. If the limit does not exist the singularity is called a ''pseudocollision'' or ''noncollision'' singularity. Paul Painlevé showed that for ''n'' = 3 any solution with a finite time singularity experiences a collision singularity. However, he failed at extending this result beyond 3 bodies. His 1895 Stockholm lectures end with the conjecture that :For ''n'' ≥ 4 the ''n''-body problem admits noncollision singularities.〔P. Painlevé, Lecons sur la théorie analytique des équations différentielles, ParisÖ Hermann, 1897〕〔Oeuvres de Paul Painlevé, Tome I, Paris Ed. Centr. Nat. Rech. Sci., 1972〕 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「Painlevé conjecture」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.