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Ermakov's Superintegrable Toy and Nonlocal Symmetries
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Date
2005-01-01
Author
Leach, P. G. L.
Karasu, Emine Ayşe
Nucci, M. C.
Andriopoulos, K.
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We investigate the symmetry properties of a pair of Ermakov equations. The system is superintegrable and yet possesses only three Lie point symmetries with the algebra sl(2, R). The number of point symmetries is insufficient and the algebra unsuitable for the complete specification of the system. We use the method of reduction of order to reduce the nonlinear fourth-order system to a third-order system comprising a linear second-order equation and a conservation law. We obtain the representation of the complete symmetry group from this system. Four of the required symmetries are nonlocal and the algebra is the direct sum of a one-dimensional Abelian algebra with the semidirect sum of a two-dimensional solvable algebra with a two-dimensional Abelian algebra. The problem illustrates the difficulties which can arise in very elementary systems. Our treatment demonstrates the existence of possible routes to overcome these problems in a systematic fashion.
Subject Keywords
Mathematical Physics
,
Geometry and Topology
,
Analysis
URI
https://hdl.handle.net/11511/34543
Journal
SYMMETRY INTEGRABILITY AND GEOMETRY-METHODS AND APPLICATIONS
DOI
https://doi.org/10.3842/sigma.2005.018
Collections
Department of Physics, Article
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P. G. L. Leach, E. A. Karasu, M. C. Nucci, and K. Andriopoulos, “Ermakov’s Superintegrable Toy and Nonlocal Symmetries,”
SYMMETRY INTEGRABILITY AND GEOMETRY-METHODS AND APPLICATIONS
, pp. 0–0, 2005, Accessed: 00, 2020. [Online]. Available: https://hdl.handle.net/11511/34543.