Rajaratnam, Krishan; McLenaghan, Raymond G.; Valero, Carlos
We review the theory of orthogonal separation of variables of the Hamilton-Jacobi equation on spaces of constant curvature, highlighting key contributions to the theory by Benenti. This theory revolves around a special type of conformal Killing tensor, hereafter called a concircular tensor. First, we show how to extend original results given by Benenti to intrinsically characterize all (orthogonal) separable coordinates in spaces of constant curvature using concircular tensors. This results in the construction of a special class of separable coordinates known as Kalnins-Eisenhart-Miller coordinates. Then we present the Benenti-Eisenhart-Kalnins-Miller separation algorithm, which uses concircular tensors to intrinsically search for Kalnins-Eisenhart-Miller coordinates which separate a given natural Hamilton-Jacobi equation. As a new application of the theory, we show how to obtain the separable coordinate systems in the two dimensional spaces of constant curvature, Minkowski and (Anti-)de Sitter space. We also apply the Benenti-Eisenhart-Kalnins-Miller separation algorithm to study the separability of the three dimensional Calogero-Moser and Morosi-Tondo systems.
location of the inhomogeneities and their geometry. Secondly we use this repr""’ntatien formula to prove a Lipschitz c,,t ,,¢us dcpcoidence estimate...A. Pericak-Spector, S. J. Spector, Nonuniqueness for a Hyperbolic System: Cavitation in Nonlinear Elastodynamics 306 E. G. Kalnins, W. Miller, Jr., q...Saturated Porous Media: Lipschitz Continuity of the Interface 336 B. J. Lucier, Regularity Through Approximation for Scalar Conservation Laws 337 B
Daskaloyannis, C. Tanoudis, Y.
The three-dimensional superintegrable systems with quadratic integrals of motion have five functionally independent integrals, one among them is the Hamiltonian. Kalnins, Kress, and Miller have proved that in the case of nondegenerate potentials with quadratic integrals of motion there is a sixth quadratic integral, which is linearly independent of the other integrals. The existence of this sixth integral implies that the integrals of motion form a ternary parafermionic-like quadratic Poisson algebra with five generators. In this contribution we investigate the structure of this algebra. We show that in all the nondegenerate cases there is at least one subalgebra of three integrals having a Poisson quadratic algebra structure, which is similar to the two-dimensional case.
Rajaratnam, Krishan McLenaghan, Raymond G.
We study Killing tensors in the context of warped products and apply the results to the problem of orthogonal separation of the Hamilton-Jacobi equation. This work is motivated primarily by the case of spaces of constant curvature where warped products are abundant. We first characterize Killing tensors which have a natural algebraic decomposition in warped products. We then apply this result to show how one can obtain the Killing-Stäckel space (KS-space) for separable coordinate systems decomposable in warped products. This result in combination with Benenti's theory for constructing the KS-space of certain special separable coordinates can be used to obtain the KS-space for all orthogonal separable coordinates found by Kalnins and Miller in Riemannian spaces of constant curvature. Next we characterize when a natural Hamiltonian is separable in coordinates decomposable in a warped product by showing that the conditions originally given by Benenti can be reduced. Finally, we use this characterization and concircular tensors (a special type of torsionless conformal Killing tensor) to develop a general algorithm to determine when a natural Hamiltonian is separable in a special class of separable coordinates which include all orthogonal separable coordinates in spaces of constant curvature.
Szereszewski, A.; Sym, A.
are binary metrics. In this paper we solve the following problem: to classify all conformally flat (of arbitrary signature) 4-dimensional binary metrics. Among them there are 1) those that are separable in the sense of SRE metrics Kalnins-Miller (1978 Trans. Am. Math. Soc. 244 241-61 1982 J. Phys. A: Math. Gen. 15 2699-709 1984 Adv. Math. 51 91-106 1983 SIAM J. Math. Anal. 14 126-37) and 2) new examples of non-Stäckel R-separability in 4 dimensions.