RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Archive Impact factor Journal history Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Fundam. Prikl. Mat.: Year: Volume: Issue: Page: Find

 Fundam. Prikl. Mat., 2015, Volume 20, Issue 2, Pages 21–34 (Mi fpm1638)

On the geometry of quadratic second-order Abel ordinary differential equations

P. V. Bibikov

Trapeznikov Institute of Control Sciences of Russian Academy of Sciences

Abstract: In this paper, we study the contact geometry of second-order ordinary differential equations that are quadratic in the highest derivative (the so-called quadratic Abel equations). Namely, we realize each quadratic Abel equation as the kernel of some nonlinear differential operator. This operator is defined by a quadratic form on the Cartan distribution in the $1$-jet space. This observation makes it possible to establish a one-to-one correspondence between quadratic Abel equations and quadratic forms on Cartan distribution. Using this realization, we construct a contact-invariant $\{e\}$-structure associated with a nondegenerate Abel equation (i.e., basis of vector fields that is invariant under contact transformations). Finally, in terms of this $\{e\}$-structure we solve the problem of contact equivalence of nondegenerate Abel equations.

 Funding Agency Grant Number Russian Foundation for Basic Research ìîë_à-14-01-31045

Full text: PDF file (176 kB)
References: PDF file   HTML file

English version:
Journal of Mathematical Sciences (New York), 2017, 223:6, 667–674

Bibliographic databases:

UDC: 517.925.4+514.763.52+514.763.8

Citation: P. V. Bibikov, “On the geometry of quadratic second-order Abel ordinary differential equations”, Fundam. Prikl. Mat., 20:2 (2015), 21–34; J. Math. Sci., 223:6 (2017), 667–674

Citation in format AMSBIB
\Bibitem{Bib15} \by P.~V.~Bibikov \paper On the geometry of quadratic second-order Abel ordinary differential equations \jour Fundam. Prikl. Mat. \yr 2015 \vol 20 \issue 2 \pages 21--34 \mathnet{http://mi.mathnet.ru/fpm1638} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3472266} \elib{http://elibrary.ru/item.asp?id=25686560} \transl \jour J. Math. Sci. \yr 2017 \vol 223 \issue 6 \pages 667--674 \crossref{https://doi.org/10.1007/s10958-017-3376-6}