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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 Non-autonomous system (mathematics) ： ウィキペディア英語版 Non-autonomous system (mathematics) In mathematics, an autonomous system is a dynamic equation on a smooth manifold. A non-autonomous system is 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-autonomous mechanics. An ''r''-order differential equation on a fiber bundle $Q\backslash to\; \backslash mathbb\; R$ is represented by a closed subbundle of a jet bundle $J^rQ$ of $Q\backslash to\; \backslash mathbb\; R$. A dynamic equation on $Q\backslash to\; \backslash mathbb\; R$ is a differential equation which is algebraically solved for a higher-order derivatives. In particular, a first-order dynamic equation on a fiber bundle $Q\backslash to\; \backslash mathbb\; R$ is a kernel of the covariant differential of some connection $\backslash Gamma$ on $Q\backslash to\; \backslash mathbb\; R$. Given bundle coordinates $(t,q^i)$ on $Q$ and the adapted coordinates $(t,q^i,q^i\_t)$ on a first-order jet manifold $J^1Q$, a first-order dynamic equation reads : $q^i\_t=\backslash Gamma\; (t,q^i).$ For instance, this is the case of Hamiltonian non-autonomous mechanics. A second-order dynamic equation : $q^i\_=\backslash xi^i(t,q^j,q^j\_t)$ on $Q\backslash to\backslash mathbb\; R$ is defined as a holonomic connection $\backslash xi$ on a jet bundle $J^1Q\backslash to\backslash mathbb\; R$. This equation also is represented by a connection on an affine jet bundle $J^1Q\backslash to\; Q$. Due to the canonical imbedding $J^1Q\backslash to\; TQ$, it is equivalent to a geodesic equation on the tangent bundle $TQ$ of $Q$. A free motion equation in non-autonomous mechanics exemplifies a second-order non-autonomous dynamic equation. == References == * De Leon, M., Rodrigues, P., Methods of Differential Geometry in Analytical Mechanics (North Holland, 1989). * Giachetta, G., Mangiarotti, L., Sardanashvily, G., Geometric Formulation of Classical and Quantum Mechanics (World Scientific, 2010) ISBN 981-4313-72-6 ((arXiv: 0911.0411 )). 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「Non-autonomous 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.