scholarly journals Second-order superintegrable systems in conformally flat spaces. I. Two-dimensional classical structure theory

2005 ◽  
Vol 46 (5) ◽  
pp. 053509 ◽  
Author(s):  
E. G. Kalnins ◽  
J. M. Kress ◽  
W. Miller
Keyword(s):  
Second Order ◽  
Structure Theory ◽  
Two Dimensional ◽  
Conformally Flat ◽  
Classical Structure ◽  
Superintegrable Systems

scholarly journals Toward Classification of 2nd Order Superintegrable Systems in 3-Dimensional Conformally Flat Spaces with Functionally Linearly Dependent Symmetry Operators

2020 ◽  
Author(s):  
Bjorn K. Berntson ◽  
◽  
Ernest G. Kalnins ◽  
Willard Miller ◽  
◽  
...  
Keyword(s):  
Euclidean Space ◽  
Flat Space ◽  
Structure Theory ◽  
Conformally Flat ◽  
3 Dimensional ◽  
Superintegrable Systems ◽  
Complex Euclidean Space ◽  
Spatial Coordinates ◽  
Symmetry Generators

We make significant progress toward the classification of 2nd order superintegrable systems on 3-dimensional conformally flat space that have functionally linearly dependent (FLD) symmetry generators, with special emphasis on complex Euclidean space. The symmetries for these systems are linearly dependent only when the coefficients are allowed to depend on the spatial coordinates. The Calogero-Moser system with 3 bodies on a line and 2-parameter rational potential is the best known example of an FLD superintegrable system. We work out the structure theory for these FLD systems on 3D conformally flat space and show, for example, that they always admit a 1st order symmetry. A partial classification of FLD systems on complex 3D Euclidean space is given. This is part of a project to classify all 3D 2nd order superintegrable systems on conformally flat spaces.

Integrable two-dimensional dynamical systems and the characteristic Lie algebras

2007 ◽  
Vol 5 ◽  
pp. 195-200
Author(s):  
A.V. Zhiber ◽  
O.S. Kostrigina
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Second Order ◽  
System Of Equations ◽  
Two Dimensional ◽  
Finite Dimensional ◽  
Dimensional Dynamical System

In the paper it is shown that the two-dimensional dynamical system of equations is Darboux integrable if and only if its characteristic Lie algebra is finite-dimensional. The class of systems having a full set of fist and second order integrals is described.

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