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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 Hamiltonian optics ： ウィキペディア英語版 Hamiltonian optics Lagrangian optics〔Vasudevan Lakshminarayanan et al., ''Lagrangian Optics'', Springer Netherlands, 2011 (978-0792375821 )〕 and Hamiltonian optics〔H. A. Buchdahl, ''An Introduction to Hamiltonian Optics'', Dover Publications, 1993 (978-0486675978 )〕 are two formulations of geometrical optics which share much of the mathematical formalism with Lagrangian mechanics and Hamiltonian mechanics. ==Hamilton's principle== (詳細はphysics, Hamilton's principle states that the evolution of a system $\backslash left(q\_1\backslash left(\backslash sigma\backslash right),\backslash cdots,q\_N\backslash left(\backslash sigma\backslash right)\backslash right)$ described by $N$ generalized coordinates between two specified states at two specified parameters ''σ''''A'' and ''σ''''B'' is a stationary point (a point where the variation is zero), of the action functional, or :$\backslash delta\; S=\; \backslash delta\backslash int\_\}\; L\backslash left(q\_1,\backslash cdots,q\_N,\backslash dot\_1,\backslash cdots,\backslash dot\_N,\backslash sigma\backslash right)\backslash ,\; d\backslash sigma=0$ where $\backslash dot\_k=dq\_k/d\backslash sigma$. Condition $\backslash delta\; S=0\; \backslash$ is valid if and only if the Euler-Lagrange equations are satisfied :$$ \frac - \frac\frac = 0 with $k=1,\backslash cdots,N$. The momentum is defined as :$p\_k=\backslash frac$ and the Euler-Lagrange equations can then be rewritten as :$$ \dot p_k = \frac where $\backslash dot\_k=dp\_k/d\backslash sigma$. A different approach to solving this problem consists in defining a Hamiltonian (taking a Legendre transform of the Lagrangian) as :$H\; =\; \backslash sum\_k\; p\_k\; -\; L$ for which a new set of differential equations can be derived by looking at how the total differential of the Lagrangian depends on parameter ''σ'', positions $q\_i\backslash ,$ and their derivatives $\backslash dot\; q\_i$ relative to ''σ''. This derivation is the same as in Hamiltonian mechanics, only with time ''t'' now replaced by a general parameter ''σ''. Those differential equations are the Hamilton's equations :$\backslash frac\; =-\; \backslash dot\_k\; \backslash ,,\; \backslash quad\; \backslash frac\; =\; \backslash dot\_k\; \backslash ,,\; \backslash quad\; \backslash frac\; =\; -\; \backslash ,.$ with $k=1,\backslash cdots,N$. Hamilton's equations are first-order differential equations, while Euler-Lagrange's equations are second-order. 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「Hamiltonian optics」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.