Geodesic
The topology of integrable systems with incomplete fields
Sbornik. Mathematics, Tome 205 (2014) no. 9, pp. 1264-1278 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Liouville's theorem holds for Hamiltonian systems with complete Hamiltonian fields which possess a complete involutive system of first integrals; such systems are called Liouville-integrable. In this paper integrable systems with incomplete Hamiltonian fields are investigated. It is shown that Liouville's theorem remains valid in the case of a single incomplete field, while if the number of incomplete fields is greater, a certain analogue of the theorem holds. An integrable system on the algebra $\mathfrak{sl}(3)$ is taken as an example. Bibliography: 11 titles.
Keywords: integrable systems, incomplete fields
Mots-clés : Liouville's theorem, Lie algebras. 
@article{SM_2014_205_9_a1,
     author = {K. R. Aleshkin},
     title = {The topology of integrable systems with incomplete fields},
     journal = {Sbornik. Mathematics},
     pages = {1264--1278},
     year = {2014},
     volume = {205},
     number = {9},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2014_205_9_a1/}
}
TY  - JOUR
AU  - K. R. Aleshkin
TI  - The topology of integrable systems with incomplete fields
JO  - Sbornik. Mathematics
PY  - 2014
SP  - 1264
EP  - 1278
VL  - 205
IS  - 9
UR  - http://geodesic.mathdoc.fr/item/SM_2014_205_9_a1/
LA  - en
ID  - SM_2014_205_9_a1
ER  - 
%0 Journal Article
%A K. R. Aleshkin
%T The topology of integrable systems with incomplete fields
%J Sbornik. Mathematics
%D 2014
%P 1264-1278
%V 205
%N 9
%U http://geodesic.mathdoc.fr/item/SM_2014_205_9_a1/
%G en
%F SM_2014_205_9_a1
K. R. Aleshkin. The topology of integrable systems with incomplete fields. Sbornik. Mathematics, Tome 205 (2014) no. 9, pp. 1264-1278. http://geodesic.mathdoc.fr/item/SM_2014_205_9_a1/

[1] V. I. Arnold, Mathematical methods of classical mechanics, Grad. Texts in Math., 60, Springer-Verlag, New York, 1978, xvi+462 pp. | MR | MR | Zbl | Zbl

[2] A. V. Bolsinov, A. T. Fomenko, Integrable Hamiltonian systems. Geometry, topology, classification, Chapman Hall/CRC, Boca Raton, FL, 2004, xvi+730 pp. | DOI | MR | MR | Zbl | Zbl

[3] A. S. Miščenko, A. T. Fomenko, “The integration of Euler equations on a semisimple Lie algebra”, Soviet Math. Dokl., 17 (1976):6 (1977), 1591–1594 | MR | Zbl

[4] A. S. Miščenko, A. T. Fomenko, “Euler equations on finite-dimensional Lie groups”, Math. USSR-Izv., 12:2 (1978), 371–389 | DOI | MR | Zbl

[5] A. T. Fomenko, A. Yu. Konyaev, “Algebra and geometry through hamiltonian systems”, Continuous and distributed systems. Theory and applications, Solid Mech. Appl., 211, Springer, Cham, 2014, 3–21 | DOI

[6] A. T. Fomenko, A. Yu. Konyaev, “New approach to symmetries and singularities in integrable Hamiltonian systems”, Topology Appl., 159:7 (2012), 1964–1975 | DOI | MR | Zbl

[7] T. A. Lepskii, “Incomplete integrable Hamiltonian systems with complex polynomial Hamiltonian of small degree”, Sb. Math., 201:10 (2010), 1511–1538 | DOI | DOI | MR | Zbl

[8] V. V. Trofimov, A. T. Fomenko, “Liouville integrability of Hamiltonian systems on Lie algebras”, Russian Math. Surveys, 39:2 (1984), 1–67 | DOI | MR | Zbl

[9] A. V. Bolsinov, A. M. Izosimov, A. Yu. Konyaev, A. A. Oshemkov, “Algebra i topologiya integriruemykh sistem. Zadachi dlya issledovaniya”, Trudy seminara po vektornomu i tenzornomu analizu s ikh prilozheniyami k geometrii, mekhanike i fizike, 28, Izd-vo Mosk. un-ta, M., 2012, 119–191

[10] S. T. Sadetov, “Dokazatelstvo gipotezy Mischenko–Fomenko”, Dokl. RAN, 397:6 (2004), 751–754 | MR

[11] B. Kostant, N. Wallach, “Gelfand–Zeitlin theory from the perspective of classical mechanics. I”, Studies in Lie theory, Progr. Math., 243, Birkhäuser, Boston, MA, 2006, 319–364 ; 2004, arXiv: math/0408342 | MR