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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 mechanics ： ウィキペディア英語版 Non-autonomous mechanics Non-autonomous mechanics describe non-relativistic mechanical systems subject to time-dependent transformations. In particular, this is the case of mechanical systems whose Lagrangians and Hamiltonians depend on the time. The configuration space of non-autonomous mechanics is a fiber bundle $Q\backslash to\; \backslash mathbb\; R$ over the time axis $\backslash mathbb\; R$ coordinated by $(t,q^i)$. This bundle is trivial, but its different trivializations $Q=\backslash mathbb\; R\backslash times\; M$ correspond to the choice of different non-relativistic reference frames. Such a reference frame also is represented by a connection $\backslash Gamma$ on $Q\backslash to\backslash mathbb\; R$ which takes a form $\backslash Gamma^i\; =0$ with respect to this trivialization. The corresponding covariant differential $(q^i\_t-\backslash Gamma^i)\backslash partial\_i$ determines the relative velocity with respect to a reference frame $\backslash Gamma$. As a consequence, non-autonomous mechanics (in particular, non-autonomous Hamiltonian mechanics) can be formulated as a covariant classical field theory (in particular covariant Hamiltonian field theory) on $X=\backslash mathbb\; R$. Accordingly, the velocity phase space of non-autonomous mechanics is the jet manifold $J^1Q$ of $Q\backslash to\; \backslash mathbb\; R$ provided with the coordinates $(t,q^i,q^i\_t)$. Its momentum phase space is the vertical cotangent bundle $VQ$ of $Q\backslash to\; \backslash mathbb\; R$ coordinated by $(t,q^i,p\_i)$ and endowed with the canonical Poisson structure. The dynamics of Hamiltonian non-autonomous mechanics is defined by a Hamiltonian form $p\_idq^i-H(t,q^i,p\_i)dt$. One can associate to any Hamiltonian non-autonomous system an equivalent Hamiltonian autonomous system on the cotangent bundle $TQ$ of $Q$ coordinated by $(t,q^i,p,p\_i)$ and provided with the canonical symplectic form; its Hamiltonian is $p-H$. == References == * De Leon, M., Rodrigues, P., Methods of Differential Geometry in Analytical Mechanics (North Holland, 1989). * Echeverria Enriquez, A., Munoz Lecanda, M., Roman Roy, N., Geometrical setting of time-dependent regular systems. Alternative models, Rev. Math. Phys. 3 (1991) 301. * Carinena, J., Fernandez-Nunez, J., Geometric theory of time-dependent singular Lagrangians, Fortschr. Phys., 41 (1993) 517. * Mangiarotti, L., Sardanashvily, G., Gauge Mechanics (World Scientific, 1998) ISBN 981-02-3603-4. * 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 mechanics」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.