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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 Appell's equation of motion ： ウィキペディア英語版 Appell's equation of motion In classical mechanics, Appell's equation of motion is an alternative general formulation of classical mechanics described by Paul Émile Appell in 1900 :$Q\_\; =\; \backslash frac$ is an arbitrary generalized acceleration, the second time derivative of the generalized coordinates ''qr'' and ''Qr'' is its corresponding generalized force; that is, the work done is given by :$dW\; =\; \backslash sum\_^\; Q\_\; dq\_$ where the index ''r'' runs over the ''D'' generalized coordinates ''qr'', which usually correspond to the degrees of freedom of the system. The function ''S'' is defined as the mass-weighted sum of the particle accelerations squared, :$S\; =\; \backslash frac\; \backslash sum\_^\; m\_\; \backslash mathbf\_^\backslash ,,$ where the index ''k'' runs over the ''N'' particles, and :$\backslash mathbf\_k\; =\; \backslash ddot\_k\}$ is the acceleration of the ''k''th particle, the second time derivative of its position vector r''k''. Each r''k'' is expressed in terms of generalized coordinates, and a''k'' is expressed in terms of the generalized accelerations. Appells formulation does not introduce any new physics to classical mechanics. It is fully equivalent to the other formulations of classical mechanics such as Newton's second law, Lagrangian mechanics, Hamiltonian mechanics, and the principle of least action. Appell's equation of motion may be more convenient in some cases, particularly when nonholonomic constraints are involved. Appell’s formulation is an application of Gauss' principle of least constraint. ==Derivation== The change in the particle positions r''k'' for an infinitesimal change in the ''D'' generalized coordinates is :$$ d\mathbf_ = \sum_^ dq_ \frac}_}_}^ Q_ dq_ = \sum_^ \mathbf_ \cdot d\mathbf_ = \sum_^ m_ \mathbf_ \cdot d\mathbf_ where Newton's second law for the ''k''th particle :$\backslash mathbf\_k\; =\; m\_k\backslash mathbf\_k$ has been used. Substituting the formula for ''d''r''k'' and swapping the order of the two summations yields the formulae :$$ dW = \sum_^ Q_ dq_ = \sum_^ m_ \mathbf_ \cdot \sum_^ dq_ \left( \frac}^ dq_ \sum_^ m_ \mathbf_ \cdot \left( \frac} = \sum_^ m_ \mathbf_ \cdot \left( \frac}^ m_ \mathbf_ \cdot \left( \frac} \sum_^ m_ \left| \mathbf_ \right|^ = \sum_^ m_ \mathbf_ \cdot \left( \frac} 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「Appell's equation of motion」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.