Algebraic linearization criteria for systems of ordinary differential equations

Abstract

Algebraic linearization criteria by means of general point transformations for systems of two second-order nonlinear ordinary differential equations (ODEs) are revisited. In previous work due to Wafo Soh and Mahomed (Int. J. Non-Linear Mech. 36:671, 2001) two four-dimensional Lie algebras that result in linearizability in terms of arbitrary point transformation for such systems were studied. Here we consider three more algebras of dimension four that result in linearization. Therefore our results supplement those of Wafo Soh and Mahomed (Int. J. Non-Linear Mech. 36:671, 2001). Moreover, it is shown that these are the only other possibilities for dimension four. Hence we provide the complete algebraic linearization criteria for dimension four algebras. Necessary and sufficient conditions for linearization via invertible maps of a nonlinear to a linear system are given. These are shown to be built up from the Lie algebraic criteria for linearization of scalar second-order ODEs. These results together with very recent work (Bagderina in J. Phys. A, Math. Theor. 43:465201, 2010) give a complete picture on linearizability properties via general point transformations for systems of two second-order ODEs. Furthermore, we provide natural extensions of these algebraic criteria for linearizing arbitrary systems of nonlinear second-order ODEs by means of point transformations. We also obtain algebraic criteria for the reduction of a linear system to the simplest system. Examples from Newtonian mechanics and geodesic equations are presented to illustrate our results.