分离变量的几何理论的最新进展
Geometrical Theory of Separation of Variables, a review of recent developments
摘要：
哈密尔顿-雅可比方程的分离变量理论“根据定义”与使用特殊类型的坐标有关，例如椭圆体上的雅可比坐标或欧几里得空间中的Stäckel系统。然而，以一种与坐标无关的方式发展这个理论是可能的，也是有用的：这就是变量分离的几何理论。它涉及几何对象（如特殊子流形和叶理）以及特殊向量和张量场，如Killing向量和Killing两张量（即一阶和二阶等距），以及它们的保形扩张;二次第一积分与Killing两张量有关。近年来，可分系统提供了从不同角度研究的数学结构。我们在这里提出了一个简短的审查，这些结构和它们的应用，特别考虑到潜在的几何形状。讨论了Killing张量、二次一阶积分及其相关的二阶微分算子之间的代数关系以及可分系统逼近的某些方面。本文已作为海报在动力学日欧洲2008年，代尔夫特2008年8月25日至29日。
英文摘要：
The Separation of Variables theory for the Hamilton-Jacobi equation is 'by definition' related to the use of special kinds of coordinates, for example Jacobi coordinates on the ellipsoid or Stäckel systems in the Euclidean space. However, it is possible and useful to develop this theory in a coordinate-independent way: this is the Geometrical Theory of Separation of Variables. It involves geometrical objects (like special submanifolds and foliations) as well as special vector and tensor fields like Killing vectors and Killing two-tensors (i.e. isometries of order one and two), and their conformal extensions; quadratic first integrals are associated with the Killing two-tensors. In the recent years Separable Systems provide mathematical structures studied from different points of view. We present here a short review of some of these structures and of their applications with particular consideration to the underlying geometry. Algebraic relations among Killing tensors, quadratic first integrals or their associated second order differential operators and some aspects of approximation with separable systems are considered. This paper has been presented as a poster at Dynamics Days Europe 2008, Delft 25-29 August 2008.
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