Geodesic
On quadratic integral Poincare--Zhukovsky's equations
Russian journal of nonlinear dynamics, Tome 8 (2012) no. 3, pp. 523-540.

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

For Poincaré–Zhukovsky's equations with non-diagonal matrices in the Hamiltonian, we obtain conditions for existence of the quadratic integral $(\mathbf{YS}, \mathbf{K}) = \mathrm{const}$ and the explisit form of it. It is shown that if the integral exists, then the equations reduce to the Schottky's case.
Keywords: Poincaré–Zhukovsky's equations, quadratic integral
Mots-clés : non-diagonal matrices, Schottky's case. 
@article{ND_2012_8_3_a7,
     author = {Vladimir Yu. Ol'shanskii},
     title = {On quadratic integral {Poincare--Zhukovsky's} equations},
     journal = {Russian journal of nonlinear dynamics},
     pages = {523--540},
     publisher = {mathdoc},
     volume = {8},
     number = {3},
     year = {2012},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ND_2012_8_3_a7/}
}
TY  - JOUR
AU  - Vladimir Yu. Ol'shanskii
TI  - On quadratic integral Poincare--Zhukovsky's equations
JO  - Russian journal of nonlinear dynamics
PY  - 2012
SP  - 523
EP  - 540
VL  - 8
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/ND_2012_8_3_a7/
LA  - ru
ID  - ND_2012_8_3_a7
ER  - 
%0 Journal Article
%A Vladimir Yu. Ol'shanskii
%T On quadratic integral Poincare--Zhukovsky's equations
%J Russian journal of nonlinear dynamics
%D 2012
%P 523-540
%V 8
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/ND_2012_8_3_a7/
%G ru
%F ND_2012_8_3_a7
Vladimir Yu. Ol'shanskii. On quadratic integral Poincare--Zhukovsky's equations. Russian journal of nonlinear dynamics, Tome 8 (2012) no. 3, pp. 523-540. http://geodesic.mathdoc.fr/item/ND_2012_8_3_a7/

[1] Zhukovskii N. E., “O dvizhenii tverdogo tela, imeyuschego polosti, napolnennye odnorodnoi kapelnoi zhidkostyu”, Sobr. soch., v. 2, Gostekhizdat, M., 1948, 31–152

[2] Poincaré H., “Sur la précession des corps déformables”, Bull. Astron. Ser. 1, 27 (1910), 321–356; Пуанкаре А., Последние работы, НИЦ «РХД», Ижевск, 2001, 74–111

[3] Moiseev N. N., Rumyantsev V. V., Dinamika tel s polostyami, soderzhaschimi zhidkost, Nauka, M., 1965, 440 pp. | Zbl

[4] Borisov A. V., Mamaev I. S., Dinamika tverdogo tela, NITs «RKhD», Izhevsk, 2001, 384 pp. | MR | Zbl

[5] Schottky F., “Über das analytische Problem der Rotation eines starren Körpers in Raume von vier Dimensionen”, Sitzungsberichte der König, 13, Preussischen Akademie der Wissenschaften, 1891, 227–232; Шоттки Ф., “Об аналитической задаче вращения твердого тела в четырехмерном пространстве”, Система Клебша. Разделение переменных, явное интегрирование, Сб. ст., ред. А. В. Борисов, А. В. Цыганов, НИЦ «РХД». Институт компьютерных исследований, М.–Ижевск, 2009, 153–158

[6] Stekloff V. A., “Sur le mouvement d'un corps solide ayant une cavité de forme ellipsoïdale remple par un liquide incompressible en sur les variations des latitudes”, Ann. de la Faculte des Sciences l'Universite de Toulouse, Ser. 3, 1 (1909), 145–256 ; Стеклов В. А., Работы по механике 1902–1909 гг., Переводы с французского, ред. А. В. Борисов, И. С. Мамаев, А. В. Цыганов, НИЦ «РХД». Институт компьютерных исследований, М.–Ижевск, 2011 | MR

[7] Veselov A. P., “Ob usloviyakh integriruemosti uravnenii Eilera na $\mathrm{SO(4)}$”, Dokl. AN SSSR, 270:6 (1983), 1298–1300 | MR | Zbl

[8] Fomenko A. T., Simplekticheskaya geometriya. Metody i prilozheniya, MGU, M., 1988, 413 pp. | MR

[9] Brusentsova E. A., Pleshakov Yu. D., “Novye integriruemye sluchai v zadache Zhukovskogo–Puankare”, Dokl. RAN, 418:4 (2008), 473–476 | MR | Zbl