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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 Helmholtz theorem (classical mechanics) ： ウィキペディア英語版 Helmholtz theorem (classical mechanics) The Helmholtz theorem of classical mechanics reads as follows: Let :$H(x,p;V)=K(p)+\backslash varphi(x;V)$ be the Hamiltonian of a one-dimensional system, where :$K=\backslash frac$ is the kinetic energy and :$\backslash varphi(x;V)$ is a "U-shaped" potential energy profile which depends on a parameter $V$. Let $\backslash left\backslash langle\; \backslash cdot\; \backslash right\backslash rangle\; \_$ denote the time average. Let :$E\; =\; K\; +\; \backslash varphi,$ :$T\; =\; 2\backslash left\backslash langle\; K\backslash right\backslash rangle\; \_,$ :$P\; =\; \backslash left\backslash langle\; -\backslash frac\backslash right\backslash rangle\; \_,$ :$S(E,V)=\backslash log\; \backslash oint\; \backslash sqrt\backslash ,dx.$ 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. This in turn allows to define the "thermodynamic state" of a one-dimensional mechanical system. In particular the temperature $T$ is given by time average of the kinetic energy, and the entropy $S$ by the logarithm of the action (i.e.$\backslash oint$ dx\sqrt). The importance of this theorem has been recognized by Ludwig Boltzmann who saw how to apply it to macroscopic systems (i.e. multidimensional systems), in order to provide a mechanical foundation of equilibrium thermodynamics. This research activity was strictly related to his formulation of the ergodic hypothesis. A multidimensional version of the Helmholtz theorem, based on the ergodic theorem of George David Birkhoff is known as generalized Helmholtz theorem. 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「Helmholtz theorem (classical 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.