Geodesic
Two integrable systems with integrals of motion of degree four
Teoretičeskaâ i matematičeskaâ fizika, Tome 186 (2016) no. 3, pp. 443-455 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We discuss the possibility of using second-order Killing tensors to construct Liouville-integrable Hamiltonian systems that are not Nijenhuis integrable. As an example, we consider two Killing tensors with a nonzero Haantjes torsion that satisfy weaker geometric conditions and also three-dimensional systems corresponding to them that are integrable in Euclidean space and have two quadratic integrals of motion and one fourth-order integral in momenta.
Keywords: Hamilton–Jacobi equation, separation of variables, Killing tensor. 
@article{TMF_2016_186_3_a6,
     author = {A. V. Tsiganov},
     title = {Two integrable systems with integrals of motion of degree four},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {443--455},
     year = {2016},
     volume = {186},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2016_186_3_a6/}
}
TY  - JOUR
AU  - A. V. Tsiganov
TI  - Two integrable systems with integrals of motion of degree four
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2016
SP  - 443
EP  - 455
VL  - 186
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/TMF_2016_186_3_a6/
LA  - ru
ID  - TMF_2016_186_3_a6
ER  - 
%0 Journal Article
%A A. V. Tsiganov
%T Two integrable systems with integrals of motion of degree four
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2016
%P 443-455
%V 186
%N 3
%U http://geodesic.mathdoc.fr/item/TMF_2016_186_3_a6/
%G ru
%F TMF_2016_186_3_a6
A. V. Tsiganov. Two integrable systems with integrals of motion of degree four. Teoretičeskaâ i matematičeskaâ fizika, Tome 186 (2016) no. 3, pp. 443-455. http://geodesic.mathdoc.fr/item/TMF_2016_186_3_a6/

[1] P. Stäckel, Ph.D. Thesis, Universität Halle, 1891

[2] L. P. Eisenhart, Ann. Math., 35 (1934), 284–305 | DOI | MR | Zbl

[3] A. Nijenhuis, Indag. Math., 54 (1951), 200–212 | DOI | MR | Zbl

[4] J. Haantjes, Indag. Math., 58 (1955), 158–162 | DOI | MR | Zbl

[5] K. Schöbel, A. P. Veselov, Commun. Math. Phys., 337:3 (2015), 1255–1274 | DOI | MR | Zbl

[6] S. Benenti, J. Math. Phys., 38:12 (1987), 6578–6602 | DOI | MR

[7] S. Benenti, “Orthogonal separable dynamical systems”, Differential Geometry and Its Applications (Opava, Czechoslovakia, August 24–28, 1992), Mathematical Publications (Opava), 1, eds. O. Kowalski, D. Krupka, Silesian Univ. Opava, Opava, 1993, 163–184 | MR | Zbl

[8] E. G. Kalnins, W. Miller, Jr., SIAM J. Math. Anal., 11:6 (1980), 1011–1026 | DOI | MR | Zbl

[9] E. G. Kalnins, W. Miller, Jr., J. Math. Phys., 27:7 (1986), 1721–1736 | DOI | MR | Zbl

[10] A. Nijenhuis, R. W. Richardson, Jr., J. Math. Mech., 17 (1967), 89–105 | MR | Zbl

[11] O. I. Bogoyavlenskii, Izv. RAN. Ser. matem., 68:6 (2004), 71–84 | DOI | DOI | MR | Zbl

[12] F. Magri, J. Phys.: Conf. Ser., 482:1 (2014), 012028, 10 pp. | DOI

[13] H. Katzin, J. Levine, Tensor (N. S.), 16 (1965), 97–104 | MR | Zbl

[14] B. Dorizzi, B. Grammaticos, J. Hietarinta, A. Ramani, F. Schwarz, Phys. Lett. A, 116:9 (1986), 432–436 | DOI | MR

[15] H. Yoshida, Celestial Mech., 31:4 (1983), 363–379 | DOI | MR | Zbl

[16] A. V. Tsiganov, Regul. Chaotic Dyn., 20:1 (2015), 74–93, arXiv: 1408.1198 | DOI | MR | Zbl

[17] A. P. Sozonov, A. V. Tsyganov, TMF, 183:3 (2015), 372–387 | DOI | DOI | MR | Zbl

[18] S. Rauch-Wojciechowski, A. V. Tsiganov, J. Phys. A: Math. Gen., 29:23 (1996), 7769–7778 | DOI | MR | Zbl

[19] A. V. Tsiganov, J. Nonlin. Math. Phys., 18:2 (2011), 245–268 | DOI | MR | Zbl

[20] Yu. A. Grigoryev, A. V. Tsiganov, J. Phys. A: Math. Theor., 44:25 (2011), 255202, 9 pp., arXiv: 1012.0468 | DOI | MR | Zbl

[21] I. A. Bizyaev, A. V. Borisov, I. S. Mamaev, Regul. Chaotic Dyn., 19:3 (2014), 415–434 | DOI | MR | Zbl

[22] B. Grammaticos, B. Dorizzi, A. Ramani, J. Hietarinta, Phys. Lett. A, 109:3 (1985), 81–84 | DOI | MR