Internal Symmetries of Differential Equations

Ian Anderson, Niky Kamran and Peter J. Olver
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Ian Anderson: Utah State University, Department of Mathematics
Niky Kamran: McGill University, Department of Mathematics
Peter J. Olver: University of Maryland, Department of Mathematics

A chapter in Modern Group Analysis: Advanced Analytical and Computational Methods in Mathematical Physics, 1993, pp 7-21 from Springer

Abstract: Abstract Bäcklund’s Theorem, which characterizes contact transformations, is generalized to give an analogous characterization of “internal symmetries” of systems of differential equations. For a wide class of systems of differential equations, every internal symmetry comes from a first order generalized symmetry and, conversely, every first order generalized symmetry satisfying certain explicit contact conditions determines an internal symmetry. We analyze the contact conditions in detail, deducing powerful necessary conditions for a system of differential equations admit “genuine” internal symmetries, i.e., ones which do not come from classical “external” symmetries. Applications include a direct proof that both the internal symmetry group and the first order generalized symmetries of a remarkable differential equation due to Hilbert and Cartan are the noncompact real form of the exceptional simple Lie group G 2.

Keywords: Vector Field; Contact Condition; Normal System; Infinitesimal Generator; Internal Symmetry (search for similar items in EconPapers)
Date: 1993
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-94-011-2050-0_2

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DOI: 10.1007/978-94-011-2050-0_2

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