Linearization criteria for second order systems of ODEs obtained from geometry

Abstract:
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.