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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 Closed geodesic ： ウィキペディア英語版 Closed geodesic In differential geometry and dynamical systems, a closed geodesic on a Riemannian manifold is a geodesic that forms a simple closed curve. It may be formalized as the projection of a closed orbit of the geodesic flow on the tangent space of the manifold. ==Definition== In a Riemannian manifold (''M'',''g''), a closed geodesic is a curve $\backslash gamma:\backslash mathbb\; R\backslash rightarrow\; M$ that is a geodesic for the metric ''g'' and is periodic. Closed geodesics can be characterized by means of a variational principle. Denoting by $\backslash Lambda\; M$ the space of smooth 1-periodic curves on ''M'', closed geodesics of period 1 are precisely the critical points of the energy function $E:\backslash Lambda\; M\backslash rightarrow\backslash mathbb\; R$, defined by $E(\backslash gamma)=\backslash int\_0^1\; g\_(\backslash dot\backslash gamma(t),\backslash dot\backslash gamma(t))\backslash ,\backslash mathrmt.$ If $\backslash gamma$ is a closed geodesic of period ''p'', the reparametrized curve $t\backslash mapsto\backslash gamma(pt)$ is a closed geodesic of period 1, and therefore it is a critical point of ''E''. If $\backslash gamma$ is a critical point of ''E'', so are the reparametrized curves $\backslash gamma^m$, for each $m\backslash in\backslash mathbb\; N$, defined by $\backslash gamma^m(t):=\backslash gamma(mt)$. Thus every closed geodesic on ''M'' gives rise to an infinite sequence of critical points of the energy ''E''. 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「Closed geodesic」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.