zbMATH — the first resource for mathematics

The Hess-Appelrot system. III: Splitting of separatrices and chaos. (English) Zbl 1398.05116
Summary: We consider a special situation of the Hess-Appelrot case of the Euler-Poisson system which describes the dynamics of a rigid body about a fixed point. One has an equilibrium point of saddle type with coinciding stable and unstable invariant 2-dimensional separatrices. We show rigorously that, after a suitable perturbation of the Hess-Appelrot case, the separatrix connection is split such that only finite number of 1-dimensional homoclinic trajectories remain and that such situation leads to a chaotic dynamics with positive entropy and to the non-existence of any additional first integral.
For Part I and II, see [the last two authors, J. Geom. Mech. 4, No. 4, 443–467 (2012; Zbl 1264.05074); J. Differ. Equations 252, No. 2, 1701–1722 (2012; Zbl 1238.05151)].

05C38 Paths and cycles
15A15 Determinants, permanents, traces, other special matrix functions
05A15 Exact enumeration problems, generating functions
15A18 Eigenvalues, singular values, and eigenvectors
Full Text: DOI
[1] R. Abraham and J. E. Marsden, Foundations of Mechanics, The Benjamin/Cummings Publ. Comp., London, 1978. · Zbl 0393.70001
[2] G. G. Appelrot, The problem of motion of a rigid body about a fixed point, Uchenye Zap. Moskov. Univ. Otdel Fiz. Mat. Nauk, 11 (1894), 1-112 [Russian].
[3] G. G. Appelrot, Incompletely symmetric heavy gyroscope, Motion of a Solid Body around a Fixed Point, Izdat. AN SSSR, Moscow-Leningrad, 1940, 61-155.
[4] V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer-Verlag, New York, 1989 [Russian: Nauka, Moskva, 1974].
[5] V. I. Arnold, V. V. Kozlov and A. I. Neishtadt, Mathematical Aspects of the Mathematical and Celestial Mechanics, Encyclopaedia of Math. Sci., Dynamical Systems, 3, Springer, Berlin-Heidelberg-New York, 1988 2nd ed., 1993 [Russian: Itogi Nauki i Tekhniki, Fundamentalnye Napravlenya, Dinamicheskiye Sistemy, 3, VINITI, Moskva, 1985]. · Zbl 1105.70002
[6] A. V. Bolsinov, A. V. Borisov and I. S. Mamaev, Topology and stability of integrable systems, Russian Math. Surveys, 65 (2010), 259-317; [Russian: Uspekhi Mat. Nauk, 65 (2010), 71-132]. · Zbl 1202.37077
[7] A. V. Borisov and I. S. Mamaev, The Hess case in the dynamics of a rigid body, J. Appl. Math. Mech., 67 (2003), 227-235 [Russian: Prikl. Mat. Mekh., 67 (2003), 256-265].
[8] R. Devaney, Homoclinic orbits in Hamiltonian systems, J. Differential Equations, 21 (1976), 431-438; Transversal homoclinic orbits in integrable system, American J. Math., 100 (1978), 631-642. · Zbl 0406.58019
[9] S. A. Dovbysh, Some new dynamical effects in the perturbed Euler-Poincot problem associated with the splitting of separatrices, Prikl. Mat. Mekh., 53 (1989), 215-225 [Russian].
[10] S. A. Dovbysh, Splitting of separatrices of unstable steady rotations and the non-integrability of the perturbed Lagrange problem, Vestnik Moskov. Univers., Ser. I. Mat. Mekh., 3 (1990), 70-77 [Russian].
[11] S. A. Dovbysh, The separatrix of an unstable position of equilibrium of the Hess-Appelrot gyroscope, Prikl. Mat. Mekh., 56 (1992), 534-545 [Russian]. · Zbl 0809.70004
[12] V. Dragović; B. Gajić, An \begindocument\(L-A\)\enddocument pair for the Hess-appelrot system and a new integrable case for the Euler-Poisson equations on \begindocument\(so(4)× so(4)\)\enddocument, Roy. Soc. Edinburgh: Proc. A, 131, 845, (2001) · Zbl 1010.70004
[13] V. V. Golubev, Lectures on Integration of the Equations of Motion of a Rigid Body about a Fixed Point, State Publ. House of Theoret. Techn. Literat., Israel Program Sci. Transl., Jerusalem, 1960 [Russian: Gosud. Izdat. Tekhnik. -Teoret. Literat., Moskva, 1953]. · Zbl 0051.15103
[14] W. Hess, Über die Euler’schen Bewegungsgleichungen und über eine neue particuäre Losung des Problems der Bewegung eines starren Körpers um einen festen Punkt, (German)Math. Ann., 37 (1890), 153-181. · JFM 22.0920.01
[15] V. V. Kozlov, Non-existence of univalent integrals and ramification of solutions in the rigid body dynamics, Prikl. Mat. Mekh., 42 (1978), 400-406 [Russian].
[16] V. V. Kozlov, Methods of Qualitative Analysis in the Rigid Body Dynamics, Izdat. Moskov. Univers., Moskva, 1980 [Russian]. · Zbl 0557.70009
[17] V. V. Kozlov and D. V. Treshchev, Non-integrability of the general problem of rotation of a dynamically symmetric heavy rigid body with fixed point. Ⅰ, Vestnik Moskov. Univers., Ser. I. Mat. Mekh., (1985), 73-81; Ⅱ, Vestnik Moskov. Univers., Ser. I. Mat. Mekh., (1986), 39-44. [Russian].
[18] P. Lubowiecki; H. Żołądek, The Hess-appelrot system. ⅰ. invariant torus and its normal hyperbolicity, J. Geometric Mechanics, 4, 443, (2012) · Zbl 1264.05074
[19] P. Lubowiecki; H. Żołądek, The Hess-appelrot system. ⅱ. perturbation and limit cycles, J. Differential Equations, 252, 1701, (2012) · Zbl 1238.05151
[20] A. J. Maciejewski; M. Przybylska, Differential Galois theory approach to the non-integrability of the heavy top, Ann. Fac.Sci. Toulouse Math., 14, 123, (2005) · Zbl 1089.70002
[21] N. A. Nekrasov, Analytic investigation of a particular case of motion of a heavy rigid body about fixed point, Matem. Sbornik, 18 (1895), 162-174 [Russian].
[22] T. V. Salnikova, Non-integrability of the perturbed Lagrange problem, Vestnik Moskov. Univers., Ser. I. Mat. Mekh., (1984), 62-66 [Russian].
[23] D. V. Turaev and L. P. Shilnikov, On Hamiltonian systems with homoclinic saddle curves, Dokl. Akad. Nauk USSR, 304 (1989), 811-814 [Russian].
[24] Yu. P. Varkhalev and G. V. Gorr, Asymptotically pendular motions of the Hess-Appelrot gyroscope, Prikl. Mat. Mekh., 48 (1984), 490-493 [Russian].
[25] N. E. Zhukovski, Geometrische interpretation des Hess’schen falles der bewegung eines schweren starken korpers um eine festen punkt, Jahr. Berichte Deutschen Math. Verein., 3, 62, (1894) · JFM 25.1440.01
[26] S. L. Ziglin, Dichotomy of separatrices, bifurcation of solutions and nonexistence of an integral in the dynamics of a rigid body, Trudy Mosk. Mat. Obshch., 41 (1980), 287-303 [Russian]. · Zbl 0466.70009
[27] S. L. Ziglin, Bifurcation of solutions and non-existence of first integrals in Hamiltonian mechanics. Ⅰ, Funct. Anal. Appl., 16 (1983), 181-189; Ⅱ, Funct. Anal. Appl., 17 (1983), 6-17; [Russian: Funkts. Anal. Prilozh., 16 (1982), 30-41; 17 (1983), 8-23]. · Zbl 0524.58015
[28] H. Żołądek, The Monodromy Group, Birkhäuser, Basel, 2006.
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.