Belyaev, A. On the full list of finite-valued solutions of the Euler-Poisson equations having four first integrals. (English) Zbl 1281.70013 Math. Nachr. 285, No. 10, 1199-1229 (2012). Summary: Till now, only 13 general and particular solutions of the problem of motion of a heavy rigid body have been found from the discovery of the first one by Euler in 1758. All these solutions are finite-valued in classical Euler and Poisson’s variables, except one in the Hess case. In this paper, we establish the full list of finite-valued solutions having four first integrals. Cited in 2 Documents MSC: 70E40 Integrable cases of motion in rigid body dynamics 70E15 Free motion of a rigid body Keywords:singular foliation; asymptotics PDFBibTeX XMLCite \textit{A. Belyaev}, Math. Nachr. 285, No. 10, 1199--1229 (2012; Zbl 1281.70013) Full Text: DOI References: [1] Yu., Arhangelskiy, The Analytic Dynamics of the Solid Body (1977) [2] Belyaev, The characteristic system for the Euler-Poisson’s equations, Nonlinear Boundary problems, National Academy of Sciences of Ukraine Institute of Appl, Math. and Mech 9 pp 135– (1999) [3] Belyaev, On single-valued solutions of the Euler-Poisson’s equations, Mat, Stud 15 (1) pp 93– (2001) [4] Belyaev, The factorization of the flow defined by the Euler-Poisson’s equations, Methods Funct, Anal. Topol 7 (4) pp 18– (2001) · Zbl 0996.34073 [5] Belyaev, The asymptotics of the solutions of the Euler-Poisson equations in the singular points of the solutions, Math, Phys. Anal. Geom 8 (2) pp 128– (2001) [6] Belyaev, On solutions of the Euler-Poissons equations which are linear combinations of {\(\zeta\)}- and -functions of Weierstrass, Mat, Stud 18 (2) pp 187– (2002) · Zbl 1075.70508 [7] Belyaev, The entire solutions of the Euler-Poisson’s equations, Ukr, Math. J 56 (5) pp 677– (2004) [8] Belyaev, Analytic properties of the solutions of the Euler-Poisson equations in the Hessian case, Ukr, Math. Bull 2 (3) pp 301– (2005) · Zbl 1211.70005 [9] Belyaev, On necessary conditions of the existence of the single-valued solutions of the Euler-Poisson equations, Mat, Stud 30 (1) pp 83– (2008) · Zbl 1164.70314 [10] Bobilyov, On a partial solution of the differential equations of the heavy solid body’s rotation around a fixed point, Trudy otdeleniya phisich. nauk obsh-va lubiteley estestvo-iya 8 (2) pp 21– (1896) [11] Borisov, Modern Methods of the Theory of Integrable Systems (2003) [12] Chaplygin, A new case of the heavy solid body’s, having a fixed point, rotation, Trudy otdeleniya phisich. nauk obsh-va lubiteley estestvo-iya 10 (2) pp 32– (1901) [13] Chaplygin, A new partial solution of the problem of the heavy solid body’s motion arround a fixed point, Trudy otdeleniya phisich. nauk obsh-va lubiteley estestvo-iya 12 (1) pp 1– (1904) [14] Dockshevitch, On a partial solution of the problem of a heavy solid body’s rotation arround a fixed point, Dokl. Akad. Nauk SSSR 167 (6) pp 1251– (1966) [15] Dockshevitch, A new partial solution of the problem of the heavy solid body’s motion arround a fixed point, Mehanika tvyordogo tela 2 pp 8– (1970) [16] L. Euler Du mouvement d’un corps solide quelconcue, lorsqu’il tourne autour d’un axe mobile Mem. de l’Academie Royale des Sciences Berlin 176 227 [17] Forster, Riemannsche Flächen (1977) · doi:10.1007/978-3-642-66547-9 [18] Goryachev, A new partial solution of the problem of the heavy solid body’s motion arround a fixed point, Trudy otdeleniya phisich. nauk obsh-va lubiteley estestvo-iya 10 (1) pp 23– (1899) [19] Goryachev, On the heavy solid body’s motion arround a fixed point in the case A = B = 4C, Mat. sbornik kruzhka lubiteley mat. nauk 21 (3) pp 431– (1900) [20] Grioli, Esistenza e determinazione delle precessioni regolari dinamicamente possibili per un solido pesante asimmetrico, Ann. Mat. Pura Appl 26 (3-4) pp 271– (1947) · Zbl 0031.03803 · doi:10.1007/BF02415381 [21] Hess, Über die Eulerschen Bewegungsgleichungen und über eine neue partikuläre Lösung des Problems der Bewegung eines starren schweren Körpers um einen festen Punkt, Math, Ann 37 (2) pp 153– (1890) [22] Kowalevski, Sur le probleme de la rotation d’un corps solide autour d’un point fixe, Acta Math. 12 (1) pp 177– (1889) · JFM 21.0935.01 · doi:10.1007/BF02592182 [23] Kowalewski, Eine neue particuläre Lösung der Differenzial gleichungen der Bewegung eines schweren starren Körpers um einen festen Punkt, Math, Ann 65 pp 528– (1908) · JFM 39.0781.02 [24] Kozlov, Integrability and non-integrability in Hamiltonian mechanics, Usp. Mat. Nauk 38 (1) pp 3– (1983) · Zbl 0525.70023 [25] Lagrange, Mecanicue Analyticue (2009) [26] Lyapunov, On the property of differential equations of the problem of the heavy solid body’s, having a fixed point, motion, Soobshheniya Har’kovskogo mat. obsh-va 4 (3) pp 1123– (1894) [27] Styeklov, A case of the heavy solid body’s, having a fixed point, motion, Trudy otdeleniya phisich. nauk obsh-va lubiteley estestvo-iya 8 (2) pp 19– (1896) [28] Styeklov, A new partial solution of the differential equations of the heavy solid body’s, having a fixed point, motion, Trudy otdeleniya phisich. nauk obsh-va lubiteley estestvo-iya 10 (1) pp 1– (1899) [29] Tamura, Topology of Foliations (1976) [30] Wells, Differential Analysis on the Complex Manifolds (1973) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.