We consider differential systems obtained by coupling two Euler–Poinsot systems. The motivation to consider such systems can be traced back to the Riemann ellipsoid problem. We provide new cases for which these systems are completely integrable. We also prove that these systems either are completely integrable or have at most four functionally independent analytic first integrals.
DOI : 10.1051/cocv/2015035
Mots-clés : Polynomial first integrals, homogeneous differential systems, Riemann ellipsoid problem, Euler–Poinsot systems, complete integrability
@article{COCV_2016__22_3_872_0,
author = {Llibre, Jaume and Valls, Cl\`audia},
title = {On the polynomial integrability of a system motivated by the {Riemann} ellipsoid problem},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {872--882},
publisher = {EDP-Sciences},
volume = {22},
number = {3},
year = {2016},
doi = {10.1051/cocv/2015035},
zbl = {1346.34002},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv/2015035/}
}
TY - JOUR AU - Llibre, Jaume AU - Valls, Clàudia TI - On the polynomial integrability of a system motivated by the Riemann ellipsoid problem JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2016 SP - 872 EP - 882 VL - 22 IS - 3 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2015035/ DO - 10.1051/cocv/2015035 LA - en ID - COCV_2016__22_3_872_0 ER -
%0 Journal Article %A Llibre, Jaume %A Valls, Clàudia %T On the polynomial integrability of a system motivated by the Riemann ellipsoid problem %J ESAIM: Control, Optimisation and Calculus of Variations %D 2016 %P 872-882 %V 22 %N 3 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2015035/ %R 10.1051/cocv/2015035 %G en %F COCV_2016__22_3_872_0
Llibre, Jaume; Valls, Clàudia. On the polynomial integrability of a system motivated by the Riemann ellipsoid problem. ESAIM: Control, Optimisation and Calculus of Variations, Tome 22 (2016) no. 3, pp. 872-882. doi : 10.1051/cocv/2015035. http://www.numdam.org/articles/10.1051/cocv/2015035/
, and , The Serret-Andoyer Riemannian metric and Euler–Poinsot rigid body motion. Math. Control Relat. Fields 3 (2013) 287–302. | DOI | Zbl
, , and , Riemannian metrics on 2D-manifolds related to the Euler–Poinsot rigid body motion. ESAIM: COCV 20 (2014) 864–893. | Numdam | Zbl
S. Chandrasekhar, Ellipsoidal figures of equilibrium. Yale University Press, New Haven (1969). | Zbl
, and , Generalized rational first integrals of analytic differential systems. J. Differ. Equ. 251 (2011) 2770–2788. | DOI | Zbl
A. Goriely, Integrability and nonintegrability of dynamical systems. Vol. 19 of Adv. Ser. Nonlin. Dyn. World Sci. Publ. Co., Inc. River Edge, NJ (2001). | Zbl
Y.N. Fedorov and V.V. Kozlov, Various Aspects of -Dimensional Rigid Body Dynamics. In Dynamical Systems in Classical Mechanics, edited by V.V. Kozlov. Vol. 168 of Amer. Math. Soc. Transl. Ser. 2 (1991). | Zbl
, and , Local Darboux first integrals of analytic differential systems. Bull. Sciences Math. 138 (2014) 71–88. | DOI | Zbl
, A remark on the integration of the Eulerian equations of the dynamics of an -dimensional rigid body. Funkcional Anal. i Prilozen. V. 6 (1972) 83–84. [English transl. Funct. Anal. Appl. 10 (1977) 328–329.] | Zbl
, Integrability, nonintegrability and chaotic motions for a system motivated by the Riemann ellipsoids problem. Regul. Chaotic Dyn. 8 (2003) 349–374. | DOI | Zbl
B. Riemann, Ein Beitrag zu den Untersuchungen über die Bewegung eines flüssigen gleichartigen Ellipsoides, Aus dem neunten Bande der Ahandlungen der Königlichen Gesellshaft der Wissenshaften zu Göttingen (1861). (Transcribed by D.R. Wilkins (2000)).
, Integrable quadratic Hamiltonians with a linear Lie-Poisson bracket, Gen. Relativity Gravitation 38 (2006) 1115–1127. | DOI | Zbl
, Local first integrals for systems of differential equations. J. Phys. A 36 (2003) 12243–12253. | DOI | Zbl
Cité par Sources :
