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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 Inverse problem for Lagrangian mechanics ： ウィキペディア英語版 Inverse problem for Lagrangian mechanics In mathematics, the inverse problem for Lagrangian mechanics is the problem of determining whether a given system of ordinary differential equations can arise as the Euler–Lagrange equations for some Lagrangian function. There has been a great deal of activity in the study of this problem since the early 20th century. A notable advance in this field was a 1941 paper by the American mathematician Jesse Douglas, in which he provided necessary and sufficient conditions for the problem to have a solution; these conditions are now known as the Helmholtz conditions, after the German physicist Hermann von Helmholtz. ==Background and statement of the problem== The usual set-up of Lagrangian mechanics on ''n''-dimensional Euclidean space R''n'' is as follows. Consider a differentiable path ''u'' : () → R''n''. The action of the path ''u'', denoted ''S''(''u''), is given by :$S(u)\; =\; \backslash int\_^\; L(t,\; u(t),\; \backslash dot(t))\; \backslash ,\; \backslash mathrm\; t,$ where ''L'' is a function of time, position and velocity known as the Lagrangian. The principle of least action states that, given an initial state ''x''0 and a final state ''x''1 in R''n'', the trajectory that the system determined by ''L'' will actually follow must be a minimizer of the action functional ''S'' satisfying the boundary conditions ''u''(0) = ''x''0, ''u''(T) = ''x''1. Furthermore, the critical points (and hence minimizers) of ''S'' must satisfy the Euler–Lagrange equations for ''S'': :$\backslash frac\; t\}\; \backslash frac\}\; -\; \backslash frac\; 1\; \backslash leq\; i\; \backslash leq\; n,$ where the upper indices ''i'' denote the components of ''u'' = (''u''1, ..., ''u''''n''). In the classical case :$T(\backslash dot)\; =\; \backslash frac\; m\; |\; \backslash dot\; |^,$ :$V\; :\; (T)\; \backslash times\; \backslash mathbb^\; \backslash to\; \backslash mathbb,$ :$L(t,\; u,\; \backslash dot)\; =\; T(\backslash dot)\; -\; V(t,\; u),$ the Euler–Lagrange equations are the second-order ordinary differential equations better known as Newton's laws of motion: :$\backslash ddot^\; =\; -\; \backslash frac\; 1\; \backslash leq\; i\; \backslash leq\; n,$ :$\backslash mbox\backslash ddot\; =\; -\; \backslash nabla\_\; V(t,\; u).$ The inverse problem of Lagrangian mechanics is as follows: given a system of second-order ordinary differential equations :$\backslash ddot^\; =\; f^\; (u^,\; \backslash dot^)\; \backslash quad\; \backslash text\; 1\; \backslash leq\; i,\; j\; \backslash leq\; n,\; \backslash quad\; \backslash mbox$ that holds for times 0 ≤ ''t'' ≤ ''T'', does there exist a Lagrangian ''L'' : () × R''n'' × R''n'' → R for which these ordinary differential equations (E) are the Euler–Lagrange equations? In general, this problem is posed not on Euclidean space R''n'', but on an ''n''-dimensional manifold ''M'', and the Lagrangian is a function ''L'' : () × T''M'' → R, where T''M'' denotes the tangent bundle of ''M''. 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「Inverse problem for Lagrangian mechanics」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.