Generalization of non-Cartan Symmetries to arbitrary dimensions

TL;DR

This paper extends the concept of non-Cartan symmetries from scalar second order ODEs to systems of arbitrary dimensions, characterizing linear systems reducible to trivial equations and exploring their properties.

Contribution

It provides explicit formulas for non-Cartan symmetries in higher dimensions and clarifies their applicability only to linear systems equivalent to trivial equations.

Findings

Non-Cartan symmetries form an abelian Lie algebra.

They characterize linear systems reducible to trivial equations.

Non-Cartan property is coordinate-free.

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. %