Eduardo García-Toraño Andrés, Tom Mestdag, “Un-Reduction of Systems of Second-Order Ordinary Differential Equations”, SIGMA, 2016, Volume 12,115, 20 pp.

This article is cited in 1 paper

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

Eduardo García-Toraño Andrés^a, 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

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.

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

MSC: 34A26; 37J15; 70H33; 70G65

Received: August 12, 2016; in final form November 29, 2016; Published online December 7, 2016

Language: English

DOI: 10.3842/SIGMA.2016.115