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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 Derrick's theorem ： ウィキペディア英語版 Derrick's theorem Derrick's theorem is an argument due to a physicist G.H. Derrick which shows that stationary localized solutions to a nonlinear wave equation or nonlinear Klein–Gordon equation in spatial dimensions three and higher are unstable. ==Original argument== Derrick's paper, which was considered an obstacle to interpreting soliton-like solutions as particles, contained the following physical argument about non-existence of stable localized stationary solutions to the nonlinear wave equation :$\backslash nabla^2\; \backslash theta-\backslash frac=\backslash frac\; 1\; 2\; f\text{'}(\backslash theta),$ \qquad \theta(x,t)\in\R,\quad x\in\R^3, , now known under the name of Derrick's Theorem. (Above, $f(s)$ is a differentiable function with $f\text{'}(0)=0$.) The energy of the time-independent solution $\backslash theta(x)\backslash ,$ is given by :$$ E=\int\left() \, d^3 x. A necessary condition for the solution to be stable is $\backslash delta^2\; E\backslash ge\; 0\backslash ,$. Suppose $\backslash theta(x)\backslash ,$ is a localized solution of $\backslash delta\; E=0\backslash ,$. Define $\backslash theta\_\backslash lambda(x)=\backslash theta(\backslash lambda\; x)\backslash ,$ where $\backslash lambda$ is an arbitrary constant, and write $I\_1=\backslash int(\backslash nabla\backslash theta)^2\; d^3\; x$, $I\_2=\backslash int\; f(\backslash theta)\; d^3\; x$. Then :$$ E_\lambda =\int\left()d^3 x =I_1/\lambda +I_2/\lambda^3. Whence $$ (dE_\lambda/d\lambda)\vert_=-I_1-3 I_2=0\,, and since $I\_1>0\backslash ,$, :$$ (d^2E_\lambda/d\lambda^2)\vert_=2 I_1+12 I_2=-2 I_1\,<0. That is, for a variation corresponding to a uniform stretching of the ''particle''. Hence the solution $\backslash theta(x)\backslash ,$ is unstable. The above argument also works for $x\backslash in\backslash R^n$, $n>3\backslash ,$. 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「Derrick's 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.