Linearizations of ordinary differential equations by area preserving maps

Abstract

We clarify the class of second and third order ordinary differential equations which can be tranformed to the simplest equations $Y''=0$ and $Y'''=0$. The coordinate changes employed to transform the equations are respectively area preserving maps for second order equations and contact form preserving maps for third order equations. A geometric explanation of the results is also given by using connections and associated covariant differentials both on tangent and cotangent spaces.