Differential Galois approach to the non-integrability of the heavy top problem

Maciejewski, Andrzej J.; Przybylska, Maria

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 14 (2005) no. 1, pp. 123-160

[1] Arnold (V.I. ). - Mathematical Methods of Classical Mechanics, Graduate Texts in Mathematics, Springer-Verlag , New York (1978). | Zbl | MR

[2] Audin (M.). - Spinning tops, volume 51 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge (1996). | Zbl | MR

[3] Baider (A. ) and Churchill (R.C.). - On monodromy groups of second-order Fuchsian equations, SIAM J. Math. Anal., 21(6), p. 1642-1652 (1990 ). | Zbl | MR

[4] Baider (A. ), Churchill (R.C.), Rod (D.L.), and Singer (M.F.). - On the infinitesimal geometry of integrable systems, in Mechanics Day (Waterloo, ON, 1992), volume 7 of Fields Inst. Commun, p. 5-56, Amer. Math. Soc., Providence, RI (1996). | Zbl | MR

[5] Beukers (F. ). - Differential Galois theory, in From Number Theory to Physics (Les Houches, 1989), pages 413-439, Springer, Berlin (1992 ). | Zbl | MR

[6] Bolotin (S.V. ). - Double asymptotic trajectories and conditions for integrability of Hamiltonian systems, Vestnik Moskov. Univ. Ser. I Mat. Mekh., (1), p. 55-63 (1990). | Zbl | MR

[7] Bolotin (S.V. ). - Variational methods for constructing chaotic motions in the dynamics of a rigid body, Prikl. Mat. Mekh. , 56(2), p. 230-239 (1992). | Zbl | MR

[8] Borisov (A. ) and Mamaev ( I.). - Dynamics of rigid body., NITS Regulyarnaya i Khaoticheskaya Dinamika, Izhevsk, in Russian (2001). | Zbl

[9] Churchill ( R.C. ) and Rod (D.L.). - On the determination of Ziglin monodromy groups, SIAM J. Math. Anal., 22(6), p. 1790-1802 (1991). | Zbl | MR

[10] Churchill ( R.C. ), Rod ( D.L.) and Singer (M.F.). - Group-theoretic obstructions to integrability, Ergodic Theory Dynam. Systems , 15(1), p. 15-48 (1995). | Zbl | MR

[11] Dokshevich ( A.I. ). - Resheniya v konechnom vide uravnenii Eilera-Puassona , "Naukova Dumka", Kiev (1992). | MR

[12] Dovbysh ( S.A.). - Splitting of separatrices of unstable uniform rotations and nonintegrability of a perturbed Lagrange problem, Vestnik Moskov. Univ. Ser. I Mat. Mekh., (3) p. 70-77 (1990). | Zbl | MR

[13] Duval (A. ) and Loday-Richaud ( M.). - Kovacic's algorithm and its application to some families of special functions, Appl. Algebra Engrg. Comm. Comput., 3(3), p. 211-246 (1992). | Zbl | MR

[14] Golubev ( V.V.). - Lectures on Integration of Equations of Motion of a Rigid Body about a Fixed Point, Gosud. Isdat. Teh. Teor. Lit., Moscow, in Russian ( 1953). | Zbl | MR

[15] Goryachev ( D.I.). - New integrable cases of integrability of the Euler dynamical equations, Warsaw Univ. Izv., in Russian (1910).

[16] Kaplansky ( I. ). - An Introduction to Differential Algebra , Hermann, Paris, second edition (1976). | MR

[17] Kimura (T. ). - On Riemann's equations which are solvable by quadratures, Funkcial. Ekvac., 12, p. 269-281 (1969/1970). | Zbl | MR

[18] Kirillov ( A.A.). - Elements of the theory of representations , Springer-Verlag, Berlin (1976), translated from the Russian by Edwin Hewitt, Grundlehren der Mathematischen Wissenschaften , Band 220. | Zbl | MR

[19] Kovacic ( J.J.). - An algorithm for solving second order linear homogeneous differential equations, J. Symbolic Comput. , 2(1), p. 3-43 (1986). | Zbl | MR

[20] Kowalevski ( S.). - Sur le problème de la rotation d'un corps solide autour d'un point fixe, Acta Math., 12, p. 177-232 (1888). | JFM

[21] Kowalevski ( S.). - Sur une propriété du système d'équations différentielles qui définit la rotation d'un corps solide autour d'un point fixe, Acta Math., 14, p. 81-93 (1890). | JFM

[22] Kozlov (V.V. ). - The nonexistence of an additional analytic integral in the problem of the motion of a nonsymmetric heavy solid body around a fixed point, Vestnik Moskov. Univ. Ser. I Mat. Mekh., (1), p. 105-110 (1975). | Zbl | MR

[23] Kozlov (V.V. ). - Methods of Qualitative Analysis in the Dynamics of a Rigid Body, Moskov. Gos. Univ., Moscow (1980). | Zbl | MR

[24] Kozlov (V.V. ) and Treshchev (D.V.). - Nonintegrability of the general problem of rotation of a dynamically symmetric heavy rigid body with a fixed point. I, Vestnik Moskov. Univ. Ser. I Mat. Mekh., (6), p. 73-81 (1985). | Zbl | MR

[25] Kozlov (V.V. ) and Treshchev (D.V.). - Nonintegrability of the general problem of rotation of a dynamically symmetric heavy rigid body with a fixed point. II, Vestnik Moskov. Univ. Ser. I Mat. Mekh., (1), p. 39-44 (1986). | Zbl | MR

[26] Lagrange ( J.L. ). - Mécanique analytique, volume 2, Chez La Veuve Desaint, Libraire, Paris (1788).

[27] Leimanis ( E. ). - The General Problem of the Motion of Coupled Rigid Bodies about a Fixed Point, Springer-Verlag , Berlin (1965). | Zbl

[28] Maciejewski ( A.J.). - Non-integrability in gravitational and cosmological models. Introduction to Ziglin theory and its differential Galois extension, in A. J. Maciejewski and B. Steves, editors, The Restless Universe. Applications of Gravitational N-Body Dynamics to Planetary, Stellar and Galactic Systems, p. 361-385 (2001).

[29] Maciejewski ( A.J.). - Non-integrability of a certain problem of rotational motion of a rigid satellite, in H. Prętka-Ziomek, E. Wnuk, P. K. Seidelmann, and D. Richardson, editors, Dynamics of Natural and Artificial Celestial Bodies, p. 187-192, Kluwer Academic Publisher (2001).

[30] Maciejewski ( A.J.), Non-integrability of certain Hamiltonian systems. Applications of Morales-Ramis differential Galois extension of Ziglin theory, in T. Crespo and Z. Hajto, editors, Differential Galois Theory , volume 58, p. 139-150, Banach Center Publication, Warsaw (2002). | Zbl | MR

[31] Maciejewski ( A.J.) and Przybylska (M.). - Non-integrability of ABC flow, Phys. Lett. A, 303, p. 265-272 (2002). | Zbl | MR

[32] Maciejewski ( A.J.) and Przybylska (M.). - Non-integrability of the Suslov problem, Regul. Chaotic Dyn., 7(1), p. 73-80 (2002). | Zbl | MR

[33] Maciejewski ( A.J.) and Przybylska (M.). - Non-integrability of restricted two-body problems in constant curvature spaces, Regul. Chaotic Dyn., 8(4), p. 413-430 (2003). | Zbl | MR

[34] Maciejewski ( A.J.) and Przybylska (M.). - Non-integrability of the problem of a rigid satellite in gravitational and magnetic fields , Celestial Mech., 87(4), p. 317-351 (2003). | Zbl | MR

[35] Magid (A.R. ). - Lectures on Differential Galois Theory, volume 7 of University Lecture Series, American Mathematical Society, Providence, RI (1994). | Zbl | MR

[36] Marsden (J.E. ) and Ratiu (T.S.). - Introduction to Mechanics and Symmetry, number 17 in Texts in Applied Mathematics, Springer Verlag, New York, Berlin, Heidelberg (1994). | Zbl | MR

[37] Morales ( J.J.) and Simó (C.). - Picard-Vessiot theory and Ziglin's theorem, J. Differential Equations, 107(1), p. 140-162 (1994). | Zbl | MR

[38] Morales-Ruiz ( J.J.). - Differential Galois Theory and Non-Integrability of Hamiltonian Systems, volume 179 of Progress in Mathematics, Birkhäuser Verlag, Basel ( 1999). | Zbl | MR

[39] Morales-Ruiz ( J.J.) and Ramis (J.P.). - Galoisian obstructions to integrability of Hamiltonian systems. I, Methods Appl. Anal., 8(1), p. 33-95 (2001). | Zbl | MR

[40] Morales-Ruiz ( J.J.) and Ramis (J.P.). - Galoisian obstructions to integrability of Hamiltonian systems. II, Methods Appl. Anal., 8(1), p. 97-111 (2001). | Zbl

[41] Ramis (J.-P. ) and Martinet ( J.). - Théorie de Galois différentielle et resommation, in Computer Algebra and Differential Equations, p. 117-214, Academic Press, London (1990). | Zbl

[42] Singer (M.F. ) and Ulmer (F.). - Galois groups of second and third order linear differential equations, J. Symbolic Comput., 16(1),p. 9-36 (1993). | Zbl | MR

[43] Singer (M.F. ) and Ulmer (F.). - Necessary conditions for Liouvillian solutions of (third order) linear differential equations, Appl. Algebra Engrg. Comm. Comput., 6(1), p. 1-22 (1995). | Zbl | MR

[44] Souriau (J.-M. ). - Structure des systèmes dynamiques, Maîtrises de mathématiques, Dunod, Paris (1970). | Zbl | MR

[45] Takano (K. ) and Bannai (E.). - A global study of Jordan-Pochhammer differential equations, Funkcial. Ekvac., 19(1), p. 85-99 (1976). | Zbl | MR

[46] Ulmer (F. ) and Weil (J.-A.). - Note on Kovacic's algorithm , J. Symbolic Comput., 22(2), p. 179-200 (1996). | Zbl | MR

[47] Van Der Put (M.) and Singer (M.F.). - Galois Theory of Linear Differential Equations, volume 328 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin (2003). | Zbl | MR

[48] Whittaker ( E.T. ) and Watson (G.N.). - A Course of Modern Analysis , Cambridge University Press, London (1935). | MR | JFM

[49] Ziglin (S.L. ). - Branching of solutions and the nonexistence of an additional first integral in the problem of an asymmetric heavy rigid body in motion relative to a fixed point., Dokl. Akad. Nauk SSSR, 251(4), p. 786-790 (1980). | Zbl | MR

[50] Ziglin (S.L. ). - Branching of solutions and non-existence of first integrals in Hamiltonian mechanics. I, Functional Anal. Appl., 16, p. 181-189 (1982). | Zbl

[51] Ziglin (S.L. ). - Branching of solutions and non-existence of first integrals in Hamiltonian mechanics. II, Functional Anal. Appl., 17, p. 6-17 (1983). | Zbl

[52] Ziglin (S.L. ). - On the absence of a real-analytic first integral in some problems of dynamics, Functional Anal. Appl., 31(1), p. 3-9 (1997). | Zbl | MR