Geodesic
Rational integrals of the second degree of two-dimentional geodesic equations
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 14 (2017), pp. 33-40.

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

For projection of two-dimensional geodesic equations we consider the problem of finding integrals that are rational in generalized velocities. We obtain the conditions of the existence of integral in the form of the quotient of polynomials of the second degree when the denominator is a squared linear polynomial. In general case first condition of the existence of the rational integral of the second degree is given. Integrals in the form of the quotient of polynomials of the first, second, third and fourth degree are constructed for the simplest case of symmetric metrics.
Keywords: geodesic equations, projection, integral. 
@article{SEMR_2017_14_a56,
     author = {Yu. Yu. Bagderina},
     title = {Rational integrals of the second degree of two-dimentional geodesic equations},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
     pages = {33--40},
     publisher = {mathdoc},
     volume = {14},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SEMR_2017_14_a56/}
}
TY  - JOUR
AU  - Yu. Yu. Bagderina
TI  - Rational integrals of the second degree of two-dimentional geodesic equations
JO  - Sibirskie èlektronnye matematičeskie izvestiâ
PY  - 2017
SP  - 33
EP  - 40
VL  - 14
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SEMR_2017_14_a56/
LA  - ru
ID  - SEMR_2017_14_a56
ER  - 
%0 Journal Article
%A Yu. Yu. Bagderina
%T Rational integrals of the second degree of two-dimentional geodesic equations
%J Sibirskie èlektronnye matematičeskie izvestiâ
%D 2017
%P 33-40
%V 14
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SEMR_2017_14_a56/
%G ru
%F SEMR_2017_14_a56
Yu. Yu. Bagderina. Rational integrals of the second degree of two-dimentional geodesic equations. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 14 (2017), pp. 33-40. http://geodesic.mathdoc.fr/item/SEMR_2017_14_a56/

[1] A. V. Bolsinov, V. S. Matveev, A. T. Fomenko, “Two-dimensional Riemannian metrics with integrable geodesic flows. Local and global geometry”, Sb. Math., 189 (1998), 1441–1466 | DOI | MR | Zbl

[2] V. V. Kozlov, “On rational integrals of geodesic flows”, Regul. Chaotic Dyn., 19 (2014), 601–606 | DOI | MR | Zbl

[3] R. Bryant, M. Dunajski, M. Eastwood, “Metrisability of two-dimensional projective structures”, J. Diff. Geom., 83 (2009), 465–499 | MR

[4] B. Kruglikov, “Invariant characterization of Liouville metrics and polynomial integrals”, J. Geom. Phys., 58 (2008), 979–995 | DOI | MR | Zbl

[5] Yu. Yu. Bagderina, “Invariants of a family of scalar second-order ODEs for Lie symmetries and first integrals”, J. Phys. A: Math. Theor., 49 (2016), 155202 | DOI | MR | Zbl

[6] M. V. Pavlov, S. P. Tsarev, “On local description of two-dimensional geodesic flows with a polynomial first integral”, J. Phys. A: Math. Theor., 49 (2016), 175201 | DOI | MR | Zbl