An L–A pair for the Hess–Apel'rot system and a new integrable case for the Euler–Poisson equations on so(4) × so(4)

Vladimir Dragović; Borislav Gajić

doi:10.1017/S0308210500001141

Abstract

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We present an L–A pair for the Hess–Apel'rot case of a heavy rigid three-dimensional body. Using it, we give an algebro-geometric integration procedure. Generalizing this L–A pair, we obtain a new completely integrable case of the Euler–Poisson equations in dimension four. Explicit formulae for integrals that are in involution are given. This system is a counterexample to one of Ratiu's theorems. A corrected version of this classification theorem is proved.

Crossref Citations

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Dragović, Vladimir and Gajić, Borislav 2006. Matrix Lax Polynomials, Geometry of Prym Varieties and Systems of Hess–Appel’rot Type. Letters in Mathematical Physics, Vol. 76, Issue. 2-3, p. 163.

Dragović, Vladimir and Gajić, Borislav 2006. Systems of Hess-Appel’rot Type. Communications in Mathematical Physics, Vol. 265, Issue. 2, p. 397.

Jovanović, Božidar 2007. Partial reductions of Hamiltonian flows and Hess–Appel'rot systems onSO(n). Nonlinearity, Vol. 20, Issue. 2, p. 221.

Jovanovic, Bozidar 2008. Symmetries and integrability. Publications de l'Institut Mathematique, Vol. 84, Issue. 98, p. 1.

Dragović, V. and Gajić, B. 2009. Elliptic curves and a new construction of integrable systems. Regular and Chaotic Dynamics, Vol. 14, Issue. 4-5, p. 466.

DRAGOVIĆ, VLADIMIR GAJIĆ, BORISLAV and JOVANOVIĆ, BOŽIDAR 2009. SYSTEMS OF HESS–APPEL'ROT TYPE AND ZHUKOVSKII PROPERTY. International Journal of Geometric Methods in Modern Physics, Vol. 06, Issue. 08, p. 1253.

Belyaev, Alexandr V 2011. Asymptotic behaviour of singular points of solutions of the problem of heavyn-dimensional body motion in the Lagrange case. Sbornik: Mathematics, Vol. 202, Issue. 11, p. 1617.

Беляев, Александр Владимирович and Belyaev, Alexandr Vladimirovich 2011. Об асимптотике особых точек решений задачи о движении тяжелого $n$-мерного тела в случае Лагранжа. Математический сборник, Vol. 202, Issue. 11, p. 55.

Lubowiecki, Paweł and Żołądek, Henryk 2012. The Hess-Appelrot system. I. Invariant torus and its normal hyperbolicity. Journal of Geometric Mechanics, Vol. 4, Issue. 4, p. 443.

Dragović, Vladimir and Gajić, Borislav 2012. On the cases of Kirchhoff and Chaplygin of the Kirchhoff equations. Regular and Chaotic Dynamics, Vol. 17, Issue. 5, p. 431.

Dragović, Vladimir and Gajić, Borislav 2014. Four-dimensional generalization of the Grioli precession. Regular and Chaotic Dynamics, Vol. 19, Issue. 6, p. 656.

Belyaev, A V 2015. On the general solution of the problem of the motion of a heavy rigid body in the Hess case. Sbornik: Mathematics, Vol. 206, Issue. 5, p. 621.

Беляев, Александр Владимирович and Belyaev, Alexandr Vladimirovich 2015. Об общем решении задачи о движении тяжелого твердого тела в случае Гесса. Математический сборник, Vol. 206, Issue. 5, p. 5.

Belyaev, A V 2016. Representation of solutions to the problem of the motion of a heavy rigid body in the Kovalevskaya case in terms of Weierstrass $ \zeta$- and $ \wp$-functions and nonintegrability of the Hess case by quadratures. Sbornik: Mathematics, Vol. 207, Issue. 7, p. 889.

Belyaev, Alexandr Vladimirovich 2016. О представлении решений задачи о движении тяжелого твердого тела в случае Ковалевской в $\zeta$- и $\wp$-функциях Вейерштрасса и неинтегрируемости в квадратурах случая Гесса. Математический сборник, Vol. 207, Issue. 7, p. 3.

Dragović, Vladimir and Gajić, Borislav 2016. Some Recent Generalizations of the Classical Rigid Body Systems. Arnold Mathematical Journal, Vol. 2, Issue. 4, p. 511.

Żoła̧dek, Henryk 2019. Perturbations of the Hess–Appelrot and the Lagrange cases in the rigid body dynamics. Journal of Geometry and Physics, Vol. 142, Issue. , p. 121.

Burov, Alexander A. Guerman, Anna D. and Nikonov, Vasily I. 2020. Asymptotic Invariant Surfaces for Non-Autonomous Pendulum-Type Systems. Regular and Chaotic Dynamics, Vol. 25, Issue. 1, p. 121.

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An L–A pair for the Hess–Apel'rot system and a new integrable case for the Euler–Poisson equations on so(4) × so(4)

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