Geodesic
On the geometry of quadratic second-order Abel ordinary differential equations
Fundamentalʹnaâ i prikladnaâ matematika, Tome 20 (2015) no. 2, pp. 21-34.

Voir la notice de l'article provenant de la source Math-Net.Ru

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. 
@article{FPM_2015_20_2_a1,
     author = {P. V. Bibikov},
     title = {On the geometry of quadratic second-order {Abel} ordinary differential equations},
     journal = {Fundamentalʹna\^a i prikladna\^a matematika},
     pages = {21--34},
     publisher = {mathdoc},
     volume = {20},
     number = {2},
     year = {2015},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FPM_2015_20_2_a1/}
}
TY  - JOUR
AU  - P. V. Bibikov
TI  - On the geometry of quadratic second-order Abel ordinary differential equations
JO  - Fundamentalʹnaâ i prikladnaâ matematika
PY  - 2015
SP  - 21
EP  - 34
VL  - 20
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/FPM_2015_20_2_a1/
LA  - ru
ID  - FPM_2015_20_2_a1
ER  - 
%0 Journal Article
%A P. V. Bibikov
%T On the geometry of quadratic second-order Abel ordinary differential equations
%J Fundamentalʹnaâ i prikladnaâ matematika
%D 2015
%P 21-34
%V 20
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/FPM_2015_20_2_a1/
%G ru
%F FPM_2015_20_2_a1
P. V. Bibikov. On the geometry of quadratic second-order Abel ordinary differential equations. Fundamentalʹnaâ i prikladnaâ matematika, Tome 20 (2015) no. 2, pp. 21-34. http://geodesic.mathdoc.fr/item/FPM_2015_20_2_a1/

[1] Alekseevskii D. V., Vinogradov A. M., Lychagin V. V., Osnovnye idei i ponyatiya differentsialnoi geometrii, Itogi nauki i tekhn. Ser. Sovrem. probl. mat. Fundam. napravleniya, 28, VINITI, M., 1988 | MR | Zbl

[2] Dubrov B. M., “Ob odnom klasse kontaktnykh invariantov sistem obyknovennykh differentsialnykh uravnenii”, Izv. vyssh. uchebn. zaved. Matematika, 2006, no. 1, 76–77 | MR | Zbl

[3] Kushner A. G., Klassifikatsiya uravnenii Monzha–Ampera, Dis. $\dots$ dokt. fiz.-mat. nauk, Astrakhan, 2009

[4] Lychagin V. V., “Kontaktnaya geometriya i nelineinye differentsialnye uravneniya v chastnykh proizvodnykh vtorogo poryadka”, DAN SSSR, 238:5 (1978), 273–276 | Zbl

[5] Chen W., Li H., “The geometry of the differential equations of the fourth order under the contact transformations”, Acta Sci. Nat. Univ. Pekinensis, 35:3 (1999), 285–296 | MR | Zbl

[6] Chern S., “The geometry of the differential equation $y'''=F(x,y,y',y'')$”, Sci. Rep. Nat. Tsing Hua Univ., 4 (1950), 97–111 | MR

[7] Dubrov B., “Contact trivialization of ordinary differential equations”, Differential Geometry and Its Applications, Proc. Conf. (Opava, August 27–31, 2001), 73–84 | MR | Zbl

[8] Kruglikov B., Point classification of 2nd order ODEs: Tresse classification revisited and beyond, arXiv: 0809.4653 | MR

[9] Kushner A. G., Lychagin V. V., Rubtsov V. N., Contact Geometry and Non-Linear Differential Equations, Cambridge Univ. Press, Cambridge, 2007 | MR

[10] Lie S., “Begründung einer Invarianten-Theorie der Berührungs-Transformationen”, Math. Ann., 8 (1874), 215–303 | DOI | MR

[11] Shurygin V. V. (jr.), “On the contact equivalence problem of second-order ODEs which are quadratic with respect to the second-order derivative”, Lobachevskii J. Math., 34:3 (2013), 264–271 | DOI | MR | Zbl

[12] Sternberg S., Lectures on Differential Geometry, Prentice Hall, Englewood Cliffs, 1964 | MR | Zbl

[13] Tresse A., Détermination des invariants ponctuels de l' équation différentielle ordinaire du second ordre $y''=\omega(x,y,y')$, Leipzig, 1896

[14] Yumaguzhin V., Differential invariants of 2-order ODEs. I, arXiv: 0804.0674v1