closed geodesic について

Words near each other

Dictionary Lists

mini英和辞書

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

\gamma:\mathbb R\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

\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:\Lambda M\rightarrow\mathbb R

, defined by

E(\gamma)=\int_0^1 g_(\dot\gamma(t),\dot\gamma(t))\,\mathrmt.

\gamma

is a closed geodesic of period ''p'', the reparametrized curve

t\mapsto\gamma(pt)

is a closed geodesic of period 1, and therefore it is a critical point of ''E''. If

\gamma

is a critical point of ''E'', so are the reparametrized curves

\gamma^m

, for each

m\in\mathbb N

, defined by

\gamma^m(t):=\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