Orthogonal Separation of the Hamilton-Jacobi Equation on Spaces of Constant Curvature

Abstract

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.

References in corpus (10)

• Invariant classification of orthogonally separable Hamiltonian systems in Euclidean space
• Conformal de Rham decomposition of Riemannian manifolds
• Orthogonal Separation of the Hamilton-Jacobi Equation on Spaces of Constant Curvature
• Geometry of warped products
• Classification of Hamilton-Jacobi separation in orthogonal coordinates with diagonal curvature
• Eigenvalues of Killing Tensors and Separable Webs on Riemannian and Pseudo-Riemannian Manifolds
• Separable bi-Hamiltonian systems with quadratic in momenta first integrals
• A characterization of the extrinsic spheres in a Riemannian manifold
• Warped products and Spaces of Constant Curvature
• Canonical forms of Self-Adjoint Operators in Minkowski Space-time

Cited by in corpus (5)

• Algebra of the symmetry operators of the Klein-Gordon-Fock equation for the case when groups of motions $G_3$ act transitively on null subsurfaces of spacetime
• Algebras of integrals of motion for the Hamilton-Jacobi and Klein-Gordon-Fock equations in spacetime with a four-parameter groups of motions in the presence of an external electromagnetic field
• Orthogonal Separation of the Hamilton-Jacobi Equation on Spaces of Constant Curvature
• Classification of Hamilton-Jacobi separation in orthogonal coordinates with diagonal curvature
• Classification of the orthogonal separable webs for the Hamilton-Jacobi and Laplace-Beltrami equations on 3-dimensional Hyperbolic and de Sitter spaces