Classical Separability Theory

Abstract

As was analysed in the previous chapter, once we find separation coordinates for a Liouville integrable system, we can integrate the system by quadratures through an appropriate separation relations. The fundamental problem in the Hamilton–Jacobi method is the systematic construction of transformation from some “natural” coordinates to separation coordinates. As was demonstrated in the previous chapter, such coordinates like Cartesian, spherical or cylindrical are separation coordinates only in very special cases. In general, separation coordinates are much less obvious and completely unpredictable. So the question about the existence of a systematic method for the construction of separation coordinates is very important. Indeed, for many decades of development of the separability theory, the method did not exist. Only recently, at the end of the twentieth century, after more than 100 years of efforts, a few different constructive methods were suggested. Obviously, the knowledge of all constants of motion for a given Liouville integrable system is not enough. Some extra information is required.