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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 Ostrogradsky instability ： ウィキペディア英語版 Ostrogradsky instability In applied mathematics, the Ostrogradsky instability is a consequence of a theorem of Mikhail Ostrogradsky in classical mechanics according to which a non-degenerate Lagrangian dependent on time derivatives of higher than the first corresponds to a linearly unstable Hamiltonian associated with the Lagrangian via a Legendre transform. The Ostrogradsky instability has been proposed as an explanation as to why no differential equations of higher order than two appear to describe physical phenomena. ==Outline of proof == The main points of the proof can be made clearer by considering a one-dimensional system with a Lagrangian $L(q,,\; )$. The Euler-Lagrange equation is :$\backslash frac\; -\; \backslash frac\; \backslash frac\backslash frac$ and vice versa. Thus, $dL/d$ is a function of $$ (if it was not, the Jacobian $\backslash det(L/(d\backslash ,\; d\_j))$ would vanish, which would mean that $L$ is degenerate), meaning that we can write $q^\; =\; F(q,,\; ,\; q^)$ or, inverting, $q\; =\; G(t,\; q\_0,\; \_0,\; \_0,\; q^\_0)$. Since the evolution of $q$ depends upon four initial parameters, this means that there are four canonical coordinates. We can write those as :$Q\_1\; :\; =\; q$ :$Q\_2\; :\; =$ and by using the definition of the conjugate momentum, :$P\_1\; :\; =\; \backslash frac\; \backslash frac$ as $=\; a(Q\_1,\; Q\_2,\; P\_2)$. Note that only ''three'' arguments are needed since the Lagrangian itself only has three free parameters. By Legendre transforming, we find the Hamiltonian to be :$H\; =\; P\_1\; Q\_2\; -\; P\_2\; a(Q\_1,\; Q\_2,\; P\_2)\; -\; L$ We now notice that the Hamiltonian is linear in $P\_1$. This is Ostrogradsky's instability, and it stems from the fact that the Lagrangian depends on fewer coordinates than there are canonical coordinates (which correspond to the initial parameters needed to specify the problem). The extension to higher dimensional systems is analogous, and the extension to higher derivatives simply mean that the phase space is of even higher dimension than the configuration space, which exacerbates the instability (since the Hamiltonian is linear in even more canonical coordinates). 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「Ostrogradsky instability」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.