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.

On the algorithmic linearizability of nonlinear ordinary differential equations

“…4. Some particular results were recently obtained for the case of two secondorder ordinary differential equations (Ayub (2012); Bagderina (2010); Soh (2001); Sookmee (2011)), but algorithmic and computational aspects of this problem are still open questions.…”

Integrability of systems of ordinary differential equations via Lie point symmetries

“…Example We start with the system admitting the Lie point symmetry generators (in extended form): This system has been linearized using four‐dimensional Lie group, which is intransitive in the space of variables in Ayub et al 28 Later was solved using the approach of invariant solutions in AlKindi and Ziad 40 . Here, we intend to apply Theorem (2.2) to find its solution in order to establish the applicability of our algorithm.…”

“…A large class of dynamical systems appears as geodesic equations for which metric of the space–time serves as the Lagrangian of the system and the isometries (Killing vectors) serve as Nöether symmetries 1,19–23 . For nonlinear systems of ODEs with no Lagrangian, neither Nöether symmetries nor Killing vectors exist and many techniques were established for using Lie point symmetries in order to integrate, linearize, or reduce the order of such systems 24–30 …”

Periodic solutions from Lie symmetries for the generalized Chen-Lee-Liu equation

“…The main feature of Sophus Lie approach is that the existence of a symmetry vector for a given differential equation indicates the existence of an invariant surface which can be applied for the construction of a similarity transformation in order to simplify the differential equation under the so-called reduction process [1][2][3][4]. In addition, Lie symmetries can be used to determine algebraic equivalent systems [5] and give linearization criteria for nonlinear differential equations [6][7][8]. Furthermore, Lie symmetries can be applied to construct conservation laws [9][10][11]; to determine new solutions from old solutions [12] and many other [4].…”