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
Read and print from thousands of top scholarly journals.
Already have an account? Log in
Bookmark this article. You can see your Bookmarks on your DeepDyve Library.
To save an article, log in first, or sign up for a DeepDyve account if you don’t already have one.
Copy and paste the desired citation format or use the link below to download a file formatted for EndNote
All DeepDyve websites use cookies to improve your online experience. They were placed on your computer when you launched this website. You can change your cookie settings through your browser.