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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 Harmonic coordinates ： ウィキペディア英語版 Harmonic coordinates In Riemannian geometry, a branch of mathematics, harmonic coordinates are a coordinate system on a Riemannian manifold each of whose coordinate functions ''x''''i'' is harmonic, meaning that it satisfies Laplace's equation :$\backslash Delta\; x^i\; =\; 0.\backslash ,$ Here Δ is the Laplace–Beltrami operator. Equivalently, regarding a coordinate system as a local diffeomorphism , the coordinate system is harmonic if and only if φ is a harmonic map of Riemannian manifolds, roughly meaning that it minimizes the elastic energy of "stretching" ''M'' into R''n''. The elastic energy is expressed via the Dirichlet energy functional :$E()=\backslash int\_M\; |d\backslash varphi|^2\backslash ,dV.$ In two dimensions, harmonic coordinates have been well understood for more than a century, and are closely related to isothermal coordinates, the latter being a special case of the former. Harmonic coordinates in higher dimensions were developed initially in the context of general relativity by (see harmonic coordinate condition). They were then introduced into Riemannian geometry by and later were studied by . The essential motivation for introducing harmonic coordinate systems is that the metric tensor is especially smooth when written in these coordinate systems. Harmonic coordinates are characterized in terms of the Christoffel symbols by means of the relation :$g^\backslash Gamma\_^k\; =\; 0\backslash ,$ and indeed, for any coordinate system at all, :$\backslash Delta\; x^k\; =\; -\; g^\backslash Gamma\_^k.$ Harmonic coordinates always exist (locally), a result which follows easily from standard results on the existence and regularity of solutions of elliptic partial differential equations. In particular, the equation :$\backslash Delta\; u^j\; =\; 0\backslash ,$ has a solution in a ball around any given point ''p'', such that ''u''''j''(''p'') and $\backslash partial\; u^j/\backslash partial\; x^i(p)$ are all prescribed. The basic regularity theorem concerning the metric in harmonic coordinates is that if the components of the metric are in the Hölder space ''C''''k'',α when expressed in ''some'' coordinate system, then they are in that same Hölder space when expressed in harmonic coordinates. Harmonic mapping to generate harmonic coordinates in regions with boundary is one of the original well known methods for grid generation in the field of computational fluid dynamics. Here the goal is to find a harmonic map of a given region (in Euclidean space or in a Riemannian manifold) to a convex region (very often a rectangle or a box in the case of grid generation in Euclidean space) with the additional requirement that the boundary map should be a homeomorphism (see the works of S. S. Sritharan in the reference list below). In general relativity, harmonic coordinates are solutions of the wave equation instead of the Laplace . This is known as the harmonic coordinate condition in physics. ==References== *. * (Integration of the Field Equations of Gravitation ). * . * * *. 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「Harmonic coordinates」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.