On the polynomial integrability of a system motivated by the Riemann ellipsoid problem
ESAIM: Control, Optimisation and Calculus of Variations, Volume 22 (2016) no. 3, pp. 872-882.

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.

Received:
DOI: 10.1051/cocv/2015035
Classification: 34C05, 34A34, 34C14
Keywords: Polynomial first integrals, homogeneous differential systems, Riemann ellipsoid problem, Euler–Poinsot systems, complete integrability
Llibre, Jaume 1; Valls, Clàudia 2

1 Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain
2 Departamento de Matemática, Instituto Superior Técnico, Universidade Técnica de Lisboa, Av. Rovisco Pais 1049–001, Lisboa, Portugal
@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://archive.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://archive.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://archive.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, Volume 22 (2016) no. 3, pp. 872-882. doi : 10.1051/cocv/2015035. http://archive.numdam.org/articles/10.1051/cocv/2015035/

B. Bonnard, O. Cots and N. Shcherbakova, The Serret-Andoyer Riemannian metric and Euler–Poinsot rigid body motion. Math. Control Relat. Fields 3 (2013) 287–302. | DOI | Zbl

B. Bonnard, O. Cots, J.B. Pomet and N. Shcherbakova, 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

W. Cong, J. Llibre and X. Zhang, 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 n-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

J. Llibre, S. Walcher and X. Zhang, Local Darboux first integrals of analytic differential systems. Bull. Sciences Math. 138 (2014) 71–88. | DOI | Zbl

S.V. Manakov, A remark on the integration of the Eulerian equations of the dynamics of an n-dimensional rigid body. Funkcional Anal. i Prilozen. V. 6 (1972) 83–84. [English transl. Funct. Anal. Appl. 10 (1977) 328–329.] | Zbl

P. Negrini, 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)).

T. Wolf, Integrable quadratic Hamiltonians with a linear Lie-Poisson bracket, Gen. Relativity Gravitation 38 (2006) 1115–1127. | DOI | Zbl

X. Zhang, Local first integrals for systems of differential equations. J. Phys. A 36 (2003) 12243–12253. | DOI | Zbl

Cited by Sources: