Geometrical Theory of Separation of Variables, a review of recent developments

TL;DR

This paper reviews recent developments in the geometrical theory of separation of variables for the Hamilton-Jacobi equation, emphasizing coordinate-independent approaches and their geometric structures like Killing tensors.

Contribution

It introduces a coordinate-independent geometric framework for separation of variables, connecting algebraic and differential geometric structures in integrable systems.

Findings

Geometric objects like Killing tensors underpin separability.

Relations among quadratic first integrals are explored.

Applications to approximation methods are discussed.

Abstract

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 St\"ackel 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.