INVARIANT LINEARIZATION CRITERIA FOR SYSTEMS OF CUBICALLY NONLINEAR SECOND-ORDER ORDINARY DIFFERENTIAL EQUATIONS

摘要

Invariant linearization criteria for square systems of second-order quadratically nonlinear ordinary differentialnequations (ODEs) that can be represented as geodesic equations are extended to square systems ofnODEs cubically nonlinear in the first derivatives. It is shown that there are two branches for the linearizationnproblem via point transformations for an arbitrary system of second-order ODEs and its reduction to thensimplest system. One is when the system is at most cubic in the first derivatives. One obtains the equivalentnof the Lie conditions for such systems. We explicitly solve this branch of the linearization problem bynpoint transformations in the case of a square system of two second-order ODEs. Necessary and sufficientnconditions for linearization to the simplest system by means of point transformations are given in termsnof coefficient functions of the system of two second-order ODEs cubically nonlinear in the first derivatives.nA consequence of our geometric approach of projection is a rederivation of Lie’s linearization conditionsnfor a single second-order ODE and sheds light on more recent results for them. In particular we show herenhow one can construct point transformations for reduction to the simplest linear equation by going to thenhigher space and just utilizing the coefficients of the original ODE. We also obtain invariant criteria for thenreduction of a linear square system to the simplest system. Moreover these results contain the quadraticncase as a special case. Examples are given to illustrate our results.