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We study the geometry of real analytic second order ODEs under the local real analytic diffeomorphism of $$\mathbb {R}^2$$ 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 $$f_y+\dfrac{2}{9}f_{y^{\prime }}^{2}-\dfrac{1}{3}\mathfrak {D}(f_{y^{\prime }})$$ f y + 2 9 f y ′ 2 - 1 3 D ( f y ′ ) . In this situation we associate a local affine normal Cartan connection on the first jet $$J^{1}(\mathbb {R},\mathbb {R})$$ J 1 ( R , 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 $$f_y+\dfrac{2}{9}f_{y^{\prime }}^{2}-\dfrac{1}{3}\mathfrak {D}(f_{y^{\prime }})\not \equiv 0$$ f y + 2 9 f y ′ 2 - 1 3 D ( f y ′ ) ≢ 0 . In the latter case we give all necessary steps in order to obtain an $$e$$ e -structure on $$J^{1}(\mathbb {R},\mathbb {R})$$ J 1 ( R , R ) for a generic second order ODE equation of that type. Finally we give the method to reduce to an $$e$$ e -structure on $$J^{1}$$ J 1 when $$f_{y^\prime y^\prime y^\prime y^\prime }\not \equiv 0$$ f y ′ y ′ y ′ y ′ ≢ 0 .
Analysis and Mathematical Physics – Springer Journals
Published: Jul 29, 2014
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