Invariant classification of orthogonally separable Hamiltonian systems in Euclidean space

The term “manifold of n dimensions” in this setting describes a set of n variables that independently take on the real values from −∞ to ∞ ([12], p 116). Motivated by this idea, one can assert that Euclidean geometry of E (Euclidean space) can be completely characterized by the invariants of the Euclidean group of transformations. As is well-known, this Lie group of (orientation-preserving) isometries, denoted here by I(E), is a semi-direct product of the corresponding groups of rotations and translations. An important aspect of Euclidean geometry is the theory of orthogonal coordinate webs that originated in works of a number of eminent mathematicians of the past including Stäckel [27], Bôcher [4], Darboux [5] and Eisenhart [8] within the framework of the theory of separation of variables. Its modern developments can be found in the review by Benenti [2] and the relevant references therein. In particular, it has been shown that there exist exactly eleven orthogonal coordinate webs which afford separation of variables for the Schrödinger and Hamilton-Jacobi equations defined in E. These coordinate webs are confocal quadrics determined by the Killing tensors of valence two having orthogonally integrable (normal) eigenvectors and distinct eigenvalues. Eisenhart’s results in E were extended by Olevsky [21] to three-dimensional spaces of non-zero constant curvature, while Kalnins, Miller and