翻訳と辞書 Words near each other ・ GAUSS Srl ・ Gauss sum ・ Gauss Tower ・ Gauss' Method ・ Gauss's constant ・ Gauss's continued fraction ・ Gauss's diary ・ Gauss's inequality ・ Gauss's law ・ Gauss's law for gravity ・ Gauss's law for magnetism ・ Gauss's lemma ・ Gauss's lemma (number theory) ・ Gauss's lemma (polynomial) ・ Gauss's lemma (Riemannian geometry) ・ Gauss's principle of least constraint ・ Gauss's Pythagorean right triangle proposal ・ Gauss-Matuyama reversal ・ Gaussan ・ Gaussan Priory ・ Gaussberg ・ Gaussia ・ Gaussia (copepod) ・ Gaussia (plant) ・ Gaussia attenuata ・ Gaussia gomez-pompae ・ Gaussia maya ・ Gaussia princeps ・ Gaussia princeps (copepod) ・ Gaussia princeps (plant) Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Gauss's principle of least constraint ： ウィキペディア英語版 Gauss's principle of least constraint The principle of least constraint is another formulation of classical mechanics enunciated by Carl Friedrich Gauss in 1829. The principle of least constraint is a least squares principle stating that the true motion of a mechanical system of $N$ masses is the minimum of the quantity :$$ Z \ \stackrel\ \sum_^ m_ \left| \frac_}_} for all trajectories satisfying any imposed constraints, where $m\_$, $\backslash mathbf\_$ and $\backslash mathbf\_$ represent the mass, position and applied forces of the $\backslash mathrm\{k^\{th\}\}$ mass. Gauss's principle is equivalent to D'Alembert's principle. The principle of least constraint is qualitatively similar to Hamilton's principle, which states that the true path taken by a mechanical system is an extremum of the action. However, Gauss's principle is a true (local) ''minimal'' principle, whereas the other is an ''extremal'' principle. ==Hertz's principle of least curvature== Hertz's principle of least curvature is a special case of Gauss's principle, restricted by the two conditions that there be no applied forces and that all masses are identical. (Without loss of generality, the masses may be set equal to one.) Under these conditions, Gauss's minimized quantity can be written :$$ Z = \sum_^ \left| \frac_} The kinetic energy $T$ is also conserved under these conditions :$$ T \ \stackrel\ \frac \sum_^ \left| \frac}\right|^ Since the line element $ds^$ in the $3N$-dimensional space of the coordinates is defined :$$ ds^ \ \stackrel\ \sum_^ \left| d\mathbf_ \right|^ the conservation of energy may also be written :$$ \left( \frac \right)^ = 2T Dividing $Z$ by $2T$ yields another minimal quantity :$$ K \ \stackrel\ \sum_^ \left| \frac_} Since $\backslash sqrt$ is the local curvature of the trajectory in the $3N$-dimensional space of the coordinates, minimization of $K$ is equivalent to finding the trajectory of least curvature (a }}\ \sum_^ \left| d\mathbf_ \right|^ the conservation of energy may also be written :$$ \left( \frac \right)^ = 2T Dividing $Z$ by $2T$ yields another minimal quantity :$$ K \ \stackrel\ \sum_^ \left| \frac_} Since $\backslash sqrt$ is the local curvature of the trajectory in the $3N$-dimensional space of the coordinates, minimization of $K$ is equivalent to finding the trajectory of least curvature (a geodesic) that is consistent with the constraints. Hertz's principle is also a special case of Jacobi's formulation of the least-action principle. 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「Gauss's principle of least constraint」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.