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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 мол_а-14-01-31045


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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}


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  • Фундаментальная и прикладная математика
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