Un-Reduction of Systems of Second-Order Ordinary Differential Equations
  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. 
Eduardo Garc\'ia-Tora\~no Andr\'es and Tom Mestdag
math.DG math-ph math.MP