Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 12 (2016), 115, 20 pages      arXiv:1606.07649      https://doi.org/10.3842/SIGMA.2016.115

Un-Reduction of Systems of Second-Order Ordinary Differential Equations

Eduardo García-Toraño Andrés a and Tom Mestdag b
a) Departamento de Matemática, Universidad Nacional del Sur, CONICET, Av. Alem 1253, 8000 Bahía Blanca, Argentina
b) Department of Mathematics and Computer Science, University of Antwerp, Middelheimlaan 1, B-2020 Antwerpen, Belgium

Received August 12, 2016, in final form November 29, 2016; Published online December 07, 2016

Abstract
In this paper we consider an alternative approach to ''un-reduction''. This is the process where one associates to a Lagrangian system on a manifold a dynamical system on a principal bundle over that manifold, in such a way that solutions project. We show that, when written in terms of second-order ordinary differential equations (SODEs), one may associate to the first system a (what we have called) ''primary un-reduced SODE'', and we explain how all other un-reduced SODEs relate to it. We give examples that show that the considered procedure exceeds the realm of Lagrangian systems and that relate our results to those in the literature.

Key words: reduction; symmetry; principal connection; second-order ordinary differential equations; Lagrangian system.

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