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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 Relativistic system (mathematics) ： ウィキペディア英語版 Relativistic system (mathematics) In mathematics, a non-autonomous system of ordinary differential equations is defined to be a dynamic equation on a smooth fiber bundle $Q\backslash to\; \backslash mathbb\; R$ over $\backslash mathbb\; R$. For instance, this is the case of non-relativistic non-autonomous mechanics, but not relativistic mechanics. To describe relativistic mechanics, one should consider a system of ordinary differential equations on a smooth manifold $Q$ whose fibration over $\backslash mathbb\; R$ is not fixed. Such a system admits transformations of a coordinate $t$ on $\backslash mathbb\; R$ depending on other coordinates on $Q$. Therefore, it is called the relativistic system. In particular, Special Relativity on the Minkowski space $Q=\; \backslash mathbb\; R^4$ is of this type. Since a configuration space $Q$ of a relativistic system has no preferable fibration over $\backslash mathbb\; R$, a velocity space of relativistic system is a first order jet manifold $J^1\_1Q$ of one-dimensional submanifolds of $Q$. The notion of jets of submanifolds generalizes that of jets of sections of fiber bundles which are utilized in covariant classical field theory and non-autonomous mechanics. A first order jet bundle $J^1\_1Q\backslash to$ Q is projective and, following the terminology of Special Relativity, one can think of its fibers as being spaces of the absolute velocities of a relativistic system. Given coordinates $(q^0,\; q^i)$ on $Q$, a first order jet manifold $J^1\_1Q$ is provided with the adapted coordinates $(q^0,q^i,q^i\_0)$ possessing transition functions : $q\text{'}^0=q\text{'}^0(q^0,q^k),\; \backslash quad\; q\text{'}^i=q\text{'}^i(q^0,q^k),\; \backslash quad$ ^i_0 = \left(\frac q^j_0 + \frac \right) \left(\frac q^j_0 + \frac \right)^. The relativistic velocities of a relativistic system are represented by elements of a fibre bundle $\backslash mathbb\; R\backslash times\; TQ$, coordinated by $(\backslash tau,q^\backslash lambda,a^\backslash lambda\_\backslash tau)$, where $TQ$ is the tangent bundle of $Q$. Then a generic equation of motion of a relativistic system in terms of relativistic velocities reads : $\backslash left(\backslash frac-\; \backslash partial\_\backslash mu$ G__\tau\cdots q^}q^\mu_ q^_\tau\cdots q^q^\mu_\tau =0, : $G\_\_\backslash tau\backslash cdots\; q^$, this is an equation of a relativistic charge in the presence of an electromagnetic field. == References == * Krasil'shchik, I. S., Vinogradov, A. M., (al. ), "Symmetries and conservation laws for differential equations of mathematical physics", Amer. Math. Soc., Providence, RI, 1999, ISBN 0-8218-0958-X. * Giachetta, G., Mangiarotti, L., Sardanashvily, G., Geometric Formulation of Classical and Quantum Mechanics (World Scientific, 2010) ISBN 981-4313-72-6 ((arXiv: 1005.1212 )). 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「Relativistic system (mathematics)」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.