Invariant Classification of Orthogonally Separable Hamiltonian Systems in Euclidean Space

Abstract

Abstract. The problem of the invariant classification of the orthogonal coordinate webs defined in Euclidean space is solved within the framework of Felix Klein's Erlangen Program. The results are applied to the problem of integrability of the Calogero-Moser model.

Nondegenerate three-dimensional complex Euclidean superintegrable systems and algebraic varieties

“…Another class of 3D superintegrable systems is that for which the five functionally independent symmetries are functionally linearly dependent. This class is related to the Calogero potential [34][35][36] and necessarily leads to first order PDEs for the potential, as well as second order. 9 However, the integrability methods discussed here should be able to handle this class with no special difficulties.…”

Second order superintegrable systems in conformally flat spaces. IV. The classical 3D Stäckel transform and 3D classification theory

“…where m i = 0, see [11,23,24,25].These potentials are superintegrable on Euclidean space and the second contains 6 parameters, which exceeds the the count of 4 for nondegenerate superintegrable systems. How can this be?…”

Hamilton-Jacobi Theory and Moving Frames

“…If U is the domain where the moving frame map ψ is well-defined, then (U, (ψ, I)) is a distinguished chart for the regular foliation whose leaves are the orbits of G in M . In particular, if O is an orbit such that U ∩ O = ∅, then each connected component of (ψ(U ∩ O), I(U ∩ O)) is of the form 5) and is a plaque of the regular foliation. The range R of the action on the regular cross-section K is the entire set of orbits through K. Slicing the distinguished chart as (2.5) suggests will only locally define these orbits as plaques of the regular foliation.…”