202 research outputs found
Invariant classification of orthogonally separable Hamiltonian systems in Euclidean space
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
Concircular tensors in Spaces of Constant Curvature: With Applications to Orthogonal Separation of The Hamilton-Jacobi Equation
We study concircular tensors in spaces of constant curvature and then apply
the results obtained to the problem of the orthogonal separation of the
Hamilton-Jacobi equation on these spaces. Any coordinates which separate the
geodesic Hamilton-Jacobi equation are called separable. Specifically for spaces
of constant curvature, we obtain canonical forms of concircular tensors modulo
the action of the isometry group, we obtain the separable coordinates induced
by irreducible concircular tensors, and we obtain warped products adapted to
reducible concircular tensors. Using these results, we show how to enumerate
the isometrically inequivalent orthogonal separable coordinates, construct the
transformation from separable to Cartesian coordinates, and execute the
Benenti-Eisenhart-Kalnins-Miller (BEKM) separation algorithm for separating
natural Hamilton-Jacobi equations.Comment: Removed preamble and references to unpublished articles. Also made
some minor changes in the bod
Orthogonal Separation of the Hamilton-Jacobi Equation on Spaces of Constant Curvature
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
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