How to Find Separation Coordinates for the Hamilton–Jacobi Equation: A Criterion of Separability for Natural Hamiltonian Systems

What an Effective Criterion of Separability says about the Calogero Type Systems

“…We present here the main steps of the complete solution to the Jacobi problem, which has been given in our paper [15]. The solution has the form of an algorithm, and we will apply it to the Calogero systems of particles interacting with each other in one dimension.…”

“…Then, by rewriting it in the canonical Euclidean frame, the transformed potential satisfies one of the three canonical forms of the BD equations (2.5). Since the general solution of these systems is known [15,Theorem 5.3], it only remains to explain what this indicates for the particular potential under study.…”

“…In three dimensions, the Calogero inverse-square potential is separable in five different coordinate systems [5,4,15]. The four-dimensional variant…”

“…We note that the parameters become the same as in the original case considered in [15] by simply setting all masses equal to unity: M = (1, 1, 1) T . If one goes through all different cases corresponding to different choices of v, w, t, s (as in [15] and as in § 3.2 above), one will find that the generalized potential is separable in exactly the same coordinate systems as the original potential; the only difference being in the choice of Euclidean reference frame.…”

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AbstractIn [15] we have proved a 1-1 correspondence between all separable coordinates on R n (according to Kalnins and Miller [9]) and systems of linear PDEs for separable potentials V (q). These PDEs, after introducing parameters reflecting the freedom of choice of Euclidean reference frame, serve as an effective criterion of separability.

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