Generalization of non-Cartan Symmetries to arbitrary dimensions

Abstract

Second order scalar ordinary differential equations ({\sc ode}s) which are linearizable possess special types of symmetries. These are the only symmetries which are non fiber-preserving in the linearized form of the equation, and they are called non-Cartan symmetries and known only for scalar {\sc ode}s. We give explicit expressions of non-Cartan symmetries for systems of {\sc ode}s of arbitrary dimensions and show that they form an abelian Lie algebra. It is however shown that the natural extension of these non-Cartan symmetries to arbitrary dimensions is applicable only to the natural extension of scalar second order equations to higher dimensions, that is, to equivalence classes under point transformations of the trivial vector equation. More precisely, it is shown that non-Cartan symmetries characterize linear systems of {\sc ode}s reducible by point transformation to their trivial counterpart, and we verify that they do not characterize nonlinear systems of {\sc ode}s having this property. It is also shown amongst others that the non-Cartan property of a symmetry vector is coordinate-free. Some examples of application of these results are discussed. %