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Search Results: 1 - 10 of 193660 matches for " Roman G. Smirnov "
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The classical Bertrand-Darboux problem
Roman G. Smirnov
Mathematics , 2006,
Abstract: The well-known problem of classical mechanics considered by Bertrand (1857) and Darboux (1901) is reviewed in the context of Cartan's geometry.
Andrew Lenard: A Mystery Unraveled
Jeffery Praught,Roman G. Smirnov
Symmetry, Integrability and Geometry : Methods and Applications , 2005,
Abstract: The theory of bi-Hamiltonian systems has its roots in what is commonly referred to as the ``Lenard recursion formula''. The story about the discovery of the formula told by Andrew Lenard is the subject of this article.
A class of superintegrable systems of Calogero type
Roman G. Smirnov,Pavel Winternitz
Physics , 2006, DOI: 10.1063/1.2749406
Abstract: We show that the three body Calogero model with inverse square potentials can be interpreted as a maximally superintegrable and multiseparable system in Euclidean three-space. As such it is a special case of a family of systems involving one arbitrary function of one variable.
Covariants,joint invariants and the problem of equivalence in the invariant theory of Killing tensors defined in pseudo-Riemannian spaces of constant curvature
Roman G. Smirnov,Jin Yue
Physics , 2004, DOI: 10.1063/1.1805728
Abstract: The invariant theory of Killing tensors (ITKT) is extended by introducing the new concepts of covariants and joint invariants of (product) vector spaces of Killing tensors defined in pseudo-Riemannian spaces of constant curvature. The covariants are employed to solve the problem of classification of the orthogonal coordinate webs generated by non-trivial Killing tensors of valence two defined in the Euclidean and Minkowski planes. Illustrative examples are provided.
Andrew Lenard: A Mystery Unraveled
Jeffery Praught,Roman G. Smirnov
Mathematics , 2005, DOI: 10.3842/SIGMA.2005.005
Abstract: The theory of bi-Hamiltonian systems has its roots in what is commonly referred to as the "Lenard recursion formula". The story about the discovery of the formula told by Andrew Lenard is the subject of this article.
The Smorodinsky-Winternitz potential revisited
Roman G. Smirnov,Amelia L. Yzaguirre
Physics , 2012,
Abstract: We employ joint invariants of Killing two-tensors defined in the Euclidean plane to characterize the Smorodinsky-Winternitz potential and explain the geometric meaning of its arbitrary parameters. In addition, we verify for which values of the arbitrary parameter $k$ the Tremblay-Turbiner-Winternitz potential is multi-separable.
Hamilton-Jacobi Theory and Moving Frames
Joshua D. MacArthur,Raymond G. McLenaghan,Roman G. Smirnov
Symmetry, Integrability and Geometry : Methods and Applications , 2007,
Abstract: The interplay between the Hamilton-Jacobi theory of orthogonal separation of variables and the theory of group actions is investigated based on concrete examples.
Hamilton-Jacobi Theory and Moving Frames
Joshua D. MacArthur,Raymond G. McLenaghan,Roman G. Smirnov
Physics , 2007, DOI: 10.3842/SIGMA.2007.070
Abstract: The interplay between the Hamilton-Jacobi theory of orthogonal separation of variables and the theory of group actions is investigated based on concrete examples.
Invariant classification of orthogonally separable Hamiltonian systems in Euclidean space
Joshua T. Horwood,Raymond G. McLenaghan,Roman G. Smirnov
Mathematics , 2006, DOI: 10.1007/s00220-005-1331-8
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.
Equivalence problem for the orthogonal webs on the sphere
Caroline Cochran,Raymond G. McLenaghan,Roman G. Smirnov
Mathematics , 2010, DOI: 10.1063/1.3578773
Abstract: We solve the equivalence problem for the orthogonally separable webs on the three-sphere under the action of the isometry group. This continues a classical project initiated by Olevsky in which he solved the corresponding canonical forms problem. The solution to the equivalence problem together with the results by Olevsky forms a complete solution to the problem of orthogonal separation of variables to the Hamilton-Jacobi equation defined on the three-sphere via orthogonal separation of variables. It is based on invariant properties of the characteristic Killing two-tensors in addition to properties of the corresponding algebraic curvature tensor and the associated Ricci tensor. The result is illustrated by a non-trivial application to a natural Hamiltonian defined on the three-sphere.
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