Asghar Qadir (National University of Sciences and Technology, Rawalpindi, Pakistan)

Linearization criteria for second order systems of ODEs obtained from geometry

Lie provided criteria to determine what equations can be written, by a suitable choice of transformation, to linear equations. More specifically, he showed that a single 2nd order ODE must be (at most) cubically semi-linear to be linearizable and stated the criteria to determine linearizability. Considering quadratically semi-linear systems of equations that could be regarded as geodesic equations it has been shown that the criteria could be stated as well. Aminova and Aminov provided a method for projecting a system of geodesic equations to a (one) lower dimensional system of cubically semi-linear equations. Using this method the criteria developed for quadratically semi-linear equations can be used to develop criteria for cubically semi-linear 2nd order ODEs. Projecting a 2-d system one obtains Lie's criteria. The procedure can also be used more generally and provides an understanding of various specific results in a more general context. The geometric method also allows one to provide some (incomplete) linearization criteria for quintically semi-linear 3rd order ODEs. In this talk the geometric methods will be reviewed and their application for linearization criteria discussed.