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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 Generalized Helmholtz theorem ： ウィキペディア英語版 Generalized Helmholtz theorem The generalized Helmholtz theorem on skates is the multi-dimensional generalization of the Helmholtz theorem which is valid only in one dimension. The generalized Helmholtz theorem reads as follows. Let :$\backslash mathbf=(p\_1,p\_2,...,p\_s),$ :$\backslash mathbf=(q\_1,q\_2,...,q\_s),$ be the canonical coordinates of a ''s''-dimensional Hamiltonian system, and let :$H(\backslash mathbf,\backslash mathbf;V)=K(\backslash mathbf)+\backslash varphi(\backslash mathbf;V)$ be the Hamiltonian function, where :$K=\backslash sum\_^\backslash frac$, is the kinetic energy and :$\backslash varphi(\backslash mathbf;V)$ is the potential energy which depends on a parameter $V$. Let the hyper-surfaces of constant energy in the 2''s''-dimensional phase space of the system be metrically indecomposable and let $\backslash left\backslash langle\; \backslash cdot\; \backslash right\backslash rangle\_t$ denote time average. Define the quantities $E$, $P$, $T$, $S$, as follows: :$E\; =\; K\; +\; \backslash varphi$, :$T\; =\; \backslash frac\backslash left\backslash langle\; K\backslash right\backslash rangle\; \_$, :$P\; =\; \backslash left\backslash langle\; -\backslash frac\backslash right\backslash rangle\; \_$, :$S(E,V)\; =\; \backslash log\; \backslash int\_;V)\; \backslash leq\; E\}\; d^s\backslash mathbfd^s\; \backslash mathbf.$ Then: :$dS\; =\; \backslash frac.$ == Remarks == The thesis of this theorem of classical mechanics reads exactly as the heat theorem of thermodynamics. This fact shows that thermodynamic-like relations exist between certain mechanical quantities in multidimensional ergodic systems. This in turn allows to define the "thermodynamic state" of a multi-dimensional ergodic mechanical system, without the requirement that the system be composed of a large number of degrees of freedom. In particular the temperature $T$ is given by twice the time average of the kinetic energy per degree of freedom, and the entropy $S$ by the logarithm of the phase space volume enclosed by the constant energy surface (i.e. the so-called volume entropy). 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「Generalized Helmholtz theorem」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.