Symmetry theory and Lagrangian inverse problem for time-dependent second-order differential equations

Abstract

A set XGamma of vector fields in the evolution space E playing the role of Newtonian vector fields, with respect to a second-order equation field Gamma , is introduced and endowed with a Cinfinity (E)-module structure. A dual set M*Gamma is used for giving an answer to the Lagrangian inverse problem. The symmetry theory is also developed in this framework and, in particular, the characterisation of symmetries of Gamma in terms of the transformation properties of the Lagrangian L is also given.