Second order ODEs under area-preserving maps

Abstract

We study the geometry of real analytic second order ODEs under the local real analytic diffeomorphism of $\mathbb {R}^2$ which are area preserving, through the method of Cartan. We obtain a subdivision into three “parts”. The first one is the most symmetric case. It is characterized by the vanishing of an area-preserving relative invariant namely $f_y+\dfrac{2}{9}f_{y^{\prime }}^{2}-\dfrac{1}{3}\mathfrak {D}(f_{y^{\prime }})$. In this situation we associate a local affine normal Cartan connection on the first jet $J^{1}(\mathbb {R},\mathbb {R})$ space whose curvature contains all the area-preserving relative differential invariants, to any second order ODE under study. The second case which includes all the Painlevé transcendents is given by the ODEs for which $f_y+\dfrac{2}{9}f_{y^{\prime }}^{2}-\dfrac{1}{3}\mathfrak {D}(f_{y^{\prime }})\not \equiv 0$. In the latter case we give all necessary steps in order to obtain an $e$-structure on $J^{1}(\mathbb {R},\mathbb {R})$ for a generic second order ODE equation of that type. Finally we give the method to reduce to an $e$-structure on $J^{1}$ when $f_{y^\prime y^\prime y^\prime y^\prime }\not \equiv 0$.