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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 Averaged Lagrangian ： ウィキペディア英語版 Averaged Lagrangian In continuum mechanics, Whitham's averaged Lagrangian method – or in short Whitham's method – is used to study the Lagrangian dynamics of slowly-varying wave trains in an inhomogeneous (moving) medium. The method is applicable to both linear and non-linear systems. As a direct consequence of the averaging used in the method, wave action is a conserved property of the wave motion. In contrast, the wave energy is not necessarily conserved, due to the exchange of energy with the mean motion. However the total energy, the sum of the energies in the wave motion and the mean motion, will be conserved for a time-invariant Lagrangian. Further, the averaged Lagrangian has a strong relation to the dispersion relation of the system. The method is due to Gerald Whitham, who developed it in the 1960s. It is for instance used in the modelling of surface gravity waves on fluid interfaces, and in plasma physics. ==Resulting equations for pure wave motion== In case a Lagrangian formulation of a continuum mechanics system is available, the averaged Lagrangian methodology can be used to find approximations for the average dynamics of wave motion – and (eventually) for the interaction between the wave motion and the mean motion – assuming the envelope dynamics of the carrier waves is slowly varying. Phase averaging of the Lagrangian results in an averaged Lagrangian, which is always independent of the wave phase itself (but depends on slowly varying wave quantities like wave amplitude, frequency and wavenumber). By Noether's theorem, variation of the averaged Lagrangian $\backslash mathcal$ with respect to the invariant wave phase $\backslash theta(\backslash boldsymbol,t)$ then gives rise to a conservation law: | }} This equation states the ''conservation of wave action'' – a generalization of the concept of an adiabatic invariant to continuum mechanics – with :$\backslash mathcal\; \backslash equiv\; -\backslash frac\; =\; +\backslash frac$ and $\backslash boldsymbol\}\; =\; -\backslash frac\}$ being the wave action $\backslash mathcal$ and wave action flux $\backslash boldsymbol$ and $t$ denote space and time respectively, while $\backslash boldsymbol$ is the gradient operator. The angular frequency $\backslash omega(\backslash boldsymbol,t)$ and wavenumber $\backslash boldsymbol(\backslash boldsymbol,t)$ are defined as〔 | }} and both are assumed to be slowly varying. Due to this definition, $\backslash omega(\backslash boldsymbol,t)$ and $\backslash boldsymbol(\backslash boldsymbol,t)$ have to satisfy the consistency relations: | }} The first consistency equation is known as the conservation of wave crests, and the second states that the wavenumber field $\backslash boldsymbol(\backslash boldsymbol,t)$ is irrotational (i.e. has zero curl). 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「Averaged Lagrangian」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.