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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 Beltrami identity ： ウィキペディア英語版 Beltrami identity The Beltrami identity, named after Eugenio Beltrami, is a simplified and less general version of the Euler–Lagrange equation in the calculus of variations. The Euler–Lagrange equation serves to extremize action functionals of the form :$I()=\backslash int\_a^b\; L()\; \backslash ,\; dx\; \backslash ,\; ,$ where are constants and . For the special case of , the Euler–Lagrange equation reduces to the Beltrami identity,〔Weisstein, Eric W. ("Euler-Lagrange Differential Equation." ) From (MathWorld )--A Wolfram Web Resource. See Eq. (5).〕 where is a constant.〔Thus, the Legendre transform of the Lagrangian, the Hamiltonian, is constant on the dynamical path.〕 ==Derivation== The following derivation of the Beltrami identity〔This derivation of the Beltrami identity corresponds to the one at — Weisstein, Eric W. ("Beltrami Identity." ) From (MathWorld )--A Wolfram Web Resource.〕 starts with the Euler–Lagrange equation, :$\backslash frac\; =\backslash frac\; \backslash frac\; \backslash ,\; .$ Multiplying both sides by , :$u\text{'}\backslash frac\; =u\text{'}\backslash frac\; \backslash frac\; \backslash ,\; .$ According to the chain rule, :$=\; u\text{'}\; +\; u\text{'}\text{'}\; +\; \backslash ,\; ,$ where . Rearranging this yields :$u\text{'}\; =\; -\; u\text{'}\text{'}\; -\; \backslash ,\; .$ Thus, substituting this expression for into the second equation of this derivation, :$-\; u\text{'}\text{'}\; -\; -u\text{'}\backslash frac\; \backslash frac\; =\; 0\; \backslash ,\; .$ By the product rule, the last term is re-expressed as :$u\text{'}\backslash frac\backslash frac=\backslash frac\backslash left(\; \backslash fracu\text{'}\; \backslash right)-\backslash fracu\text{'}\text{'}\; \backslash ,\; ,$ and rearranging, :$\backslash left(\; \}\; \backslash right)\; =\; \backslash ,\; .$ For the case of , this reduces to :$\backslash left(\; \}\; \backslash right)\; =\; 0\; \backslash ,\; ,$ so that taking the antiderivative results in the Beltrami identity, :$L\; -\; u\text{'}\backslash frac\; =\; C\; \backslash ,\; ,$ where is a constant. 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「Beltrami identity」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.