Geodesic
On an integrable system on a plane with an integral of motion of sixth order in momenta
Russian journal of nonlinear dynamics, Tome 13 (2017) no. 1, pp. 117-127.

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

In the framework of the Jacobi method we obtain a new integrable system on the plane with a natural Hamilton function and a second integral of motion which is a polynomial of sixth order in momenta. The corresponding variables of separation are images of usual parabolic coordinates on the plane after a suitable Bäcklund transformation. We also present separated relations and prove that the corresponding vector field is bi-Hamiltonian.
Keywords: finite-dimensional integrable systems, separation of variables, Bäcklund transformations. 
@article{ND_2017_13_1_a7,
     author = {A. V. Tsiganov},
     title = {On an integrable system on a plane with an integral of motion of sixth order in momenta},
     journal = {Russian journal of nonlinear dynamics},
     pages = {117--127},
     publisher = {mathdoc},
     volume = {13},
     number = {1},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ND_2017_13_1_a7/}
}
TY  - JOUR
AU  - A. V. Tsiganov
TI  - On an integrable system on a plane with an integral of motion of sixth order in momenta
JO  - Russian journal of nonlinear dynamics
PY  - 2017
SP  - 117
EP  - 127
VL  - 13
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/ND_2017_13_1_a7/
LA  - ru
ID  - ND_2017_13_1_a7
ER  - 
%0 Journal Article
%A A. V. Tsiganov
%T On an integrable system on a plane with an integral of motion of sixth order in momenta
%J Russian journal of nonlinear dynamics
%D 2017
%P 117-127
%V 13
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/ND_2017_13_1_a7/
%G ru
%F ND_2017_13_1_a7
A. V. Tsiganov. On an integrable system on a plane with an integral of motion of sixth order in momenta. Russian journal of nonlinear dynamics, Tome 13 (2017) no. 1, pp. 117-127. http://geodesic.mathdoc.fr/item/ND_2017_13_1_a7/

[1] Abel N. H., “Mémoire sur une propriéeté géenéerale d'une class très éntendue des fonctions transcendantes”, ØE uvres complétes de Niels Henrik Abel, v. 1, eds. L. Sylow, S. Lie, Grondahl Son, Christiania, 1881, 145–211 | MR

[2] Baker H. F., Abelian functions: Abel's theorem and the allied theory of theta functions, Cambridge Univ. Press, Cambridge, 1996, 724 pp. | MR

[3] Green M. L., Griffiths Ph. A., “Abel's differential equations”, Houston J. Math., 28:2 (2002), 329–351 | MR | Zbl

[4] Grigoryev Yu. A., Sozonov A. P., Tsiganov A. V., “On an integrable system on the plane with velocity-dependent potential”, Nelin. Dinam., 12:3 (2016), 355–367 (Russian)

[5] Jacobi C. G. J., Vorlesungen über Dynamik, 2nd ed., Reimer, Berlin, 1884, 300 pp. | Zbl

[6] Sozonov A. P., Tsiganov A. V., “Bäcklund transformations relating different Hamilton – Jacobi equations”, Theoret. and Math. Phys., 183:3 (2015), 768–781 | DOI | DOI | MR | Zbl

[7] Suris Yu. B., The problem of integrable discretization: Hamiltonian approach, Progr. Math., 219, Birkhäuser, Boston, Mass., 2003, 1070 pp. | MR | Zbl

[8] Tsiganov A. V., “Killing tensors with nonvanishing Haantjes torsion and integrable systems”, Regul. Chaotic Dyn., 20:4 (2015), 463–475 | DOI | MR | Zbl

[9] Tsiganov A. V., “Simultaneous separation for the Neumann and Chaplygin systems”, Regul. Chaotic Dyn., 20:1 (2015), 74–93 | DOI | MR | Zbl

[10] Tsiganov A. V., “On the Chaplygin system on the sphere with velocity dependent potential”, J. Geom. Phys., 92 (2015), 94–99 | DOI | MR | Zbl

[11] Tsiganov A. V., “On auto and hetero Bäcklund transformations for the Hénon – Heiles systems”, Phys. Lett. A, 379:45–46 (2015), 2903–2907 | DOI | MR | Zbl

[12] Veselov A. P., “Integrable maps”, Russian Math. Surveys, 46:5 (1991), 1–51 | DOI | MR | Zbl

[13] Weierstrass K., “Über die geodätischen Linien auf dem dreiachsigen Ellipsoid”, Mathematische Werke, v. 1, Mayer Müller, Berlin, 1894, 257–266

[14] Wojciechowski S., “The analogue of the Bäcklund transformation for integrable many-body systems”, J. Phys. A, 15:12 (1982), L653–L657 | DOI | MR