# American Institute of Mathematical Sciences

2012, 4(4): 443-467. doi: 10.3934/jgm.2012.4.443

## The Hess-Appelrot system. I. Invariant torus and its normal hyperbolicity

 1 Institute of Mathematics, University of Warsaw, ul. Banacha 2, 02-097 Warsaw, Poland, Poland

Received  November 2011 Revised  August 2012 Published  January 2013

We study the Hess--Appelrot case of the Euler--Poisson system which describes dynamics of a rigid body about a fixed point. We prove existence of an invariant torus which supports hyperbolic or parabolic or elliptic periodic or elliptic quasi--periodic dynamics. In the elliptic cases we study the question of normal hyperbolicity of the invariant torus in the case when the torus is close to a `critical circle'. It turns out that the normal hyperbolicity takes place only in the case of $1:q$ resonance. In the sequent paper [16] we study limit cycles which appear after perturbation of the above situation.
Citation: Paweł Lubowiecki, Henryk Żołądek. The Hess-Appelrot system. I. Invariant torus and its normal hyperbolicity. Journal of Geometric Mechanics, 2012, 4 (4) : 443-467. doi: 10.3934/jgm.2012.4.443
##### References:
 [1] G. G. Appelrot, The problem of motion of a rigid body about a fixed point,, Uchenye Zap. Mosk. Univ. Otdel Fiz. Mat. Nauk, 11 (1894), 1. [2] V. I. Arnold, "Mathematical Methods of Classical Mechanics,", Springer-Verlag, (1989). [3] A. V. Bolsinov, A. V. Borisov and I. S. Mamaev, Topology and stability of integrable systems,, Russian Math. Surveys, 65 (2010), 259. doi: 10.1070/RM2010v065n02ABEH004672. [4] M. Bobieński and H. Żołądek, Limit cycles for multidimensional vector fields. The elliptic case,, J. Dynam. Control Systems, 9 (2003), 265. doi: 10.1023/A:1023241823216. [5] M. Bobieński and H. Żołądek, A counterexample to a multidimensional version of the weakened Hilbert's 16th problem,, Moscow Math. J., 7 (2007), 1. [6] A. V. Borisov and I. S. Mamaev, The Hess case in the dynamics of a rigid body,, J. Appl. Math. Mech., 67 (2003), 227. doi: 10.1016/S0021-8928(03)90009-8. [7] A. Delshamps, R. de la Llave and T. M. Seara, A geometric approach to the existence of orbits with unbounded energy in generic periodic perturbations by a potential of generic geodesic flow of $\mathbbT^{2},$, Commun. Math. Phys., 209 (2000), 353. [8] S. A. Dovbysh, The separatrix of an unstable position of equilibrium of the Hess-Appelrot gyroscope,, Prikl. Mat. Mekh., 56 (1992), 534. doi: 10.1016/0021-8928(92)90009-W. [9] V. Dragović and B. Gajić, An $L-A$ pair for the Hess-Appelrot system and a new integrable case for the Euler-Poisson equations on $so(4)\times so(4)$,, Roy. Soc. Edinburgh: Proc. A, 131 (2001), 845. doi: 10.1017/S0308210500001141. [10] V. Dragović and B. Gajić, Systems of Hess-Appel'rot type,, Commun. Math. Phys., 265 (2006), 397. [11] 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., (1960). [12] W. Hess, Über die Euler'schen Bewegungsgleichungen und über eine neue particuläre Losung des Problems der Bewegung eines starren Körpers um eined festen Punkt,, Math. Annalen, 37 (1890), 178. [13] M. W. Hirsch, C. C. Pugh and M. Shub, "Invariant Manifolds,", Lectute Notes in Math., 583 (1977). [14] G. Lamé, Sur les surfaces isothermes dans les corps homogénes en équilibre de température,, J. Math. Pures Appl., 2 (1837), 147. [15] P. Leszczyński and H. Żołądek, Limit cycles appearing after perturbation of certain multidimensional vector fields,, J. Dynam. Differ. Equations, 13 (2001), 689. [16] P. Lubowiecki and H. Żołądek, The Hess-Appelrot system. II. Perturbation and limit cycles,, J. Differential Equations, 252 (2012), 1701. doi: 10.1016/j.jde.2011.06.012. [17] A. J. Maciejewski and M. Przybylska, Differential Galois theory approach to the non-integrability of the heavy top,, Ann. Fac. Sci. Touluse, 14 (2005), 123. [18] N. A. Nekrasov, Analytic investigation of a particular case of motion of a heavy rigid body about fixed point,, Matem. Sbornik, 18 (1895), 162. [19] Yu. P. Varkhalev and G. V. Gorr, Asymptotically pendulum motions of the Hess-Appelrot gyroscope,, Prikl. Mat. Mekh., 48 (1984), 490. [20] N. E. Zhukovski, Geometrische interpretation des Hess'schen falles der bewegung eines schweren starren korpers um eine festen punkt,, Jahr. Berichte Deutschen Math. Verein., 3 (1894), 62. [21] S. L. Ziglin, Branching of solutions and non-existence of first integrals in Hamiltonian mechanics. I,, Funct. Anal. Appl., 16 (1983), 181. [22] H. Żołądek, "The Monodromy Group,", Birkhäuser, (2006).

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##### References:
 [1] G. G. Appelrot, The problem of motion of a rigid body about a fixed point,, Uchenye Zap. Mosk. Univ. Otdel Fiz. Mat. Nauk, 11 (1894), 1. [2] V. I. Arnold, "Mathematical Methods of Classical Mechanics,", Springer-Verlag, (1989). [3] A. V. Bolsinov, A. V. Borisov and I. S. Mamaev, Topology and stability of integrable systems,, Russian Math. Surveys, 65 (2010), 259. doi: 10.1070/RM2010v065n02ABEH004672. [4] M. Bobieński and H. Żołądek, Limit cycles for multidimensional vector fields. The elliptic case,, J. Dynam. Control Systems, 9 (2003), 265. doi: 10.1023/A:1023241823216. [5] M. Bobieński and H. Żołądek, A counterexample to a multidimensional version of the weakened Hilbert's 16th problem,, Moscow Math. J., 7 (2007), 1. [6] A. V. Borisov and I. S. Mamaev, The Hess case in the dynamics of a rigid body,, J. Appl. Math. Mech., 67 (2003), 227. doi: 10.1016/S0021-8928(03)90009-8. [7] A. Delshamps, R. de la Llave and T. M. Seara, A geometric approach to the existence of orbits with unbounded energy in generic periodic perturbations by a potential of generic geodesic flow of $\mathbbT^{2},$, Commun. Math. Phys., 209 (2000), 353. [8] S. A. Dovbysh, The separatrix of an unstable position of equilibrium of the Hess-Appelrot gyroscope,, Prikl. Mat. Mekh., 56 (1992), 534. doi: 10.1016/0021-8928(92)90009-W. [9] V. Dragović and B. Gajić, An $L-A$ pair for the Hess-Appelrot system and a new integrable case for the Euler-Poisson equations on $so(4)\times so(4)$,, Roy. Soc. Edinburgh: Proc. A, 131 (2001), 845. doi: 10.1017/S0308210500001141. [10] V. Dragović and B. Gajić, Systems of Hess-Appel'rot type,, Commun. Math. Phys., 265 (2006), 397. [11] 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., (1960). [12] W. Hess, Über die Euler'schen Bewegungsgleichungen und über eine neue particuläre Losung des Problems der Bewegung eines starren Körpers um eined festen Punkt,, Math. Annalen, 37 (1890), 178. [13] M. W. Hirsch, C. C. Pugh and M. Shub, "Invariant Manifolds,", Lectute Notes in Math., 583 (1977). [14] G. Lamé, Sur les surfaces isothermes dans les corps homogénes en équilibre de température,, J. Math. Pures Appl., 2 (1837), 147. [15] P. Leszczyński and H. Żołądek, Limit cycles appearing after perturbation of certain multidimensional vector fields,, J. Dynam. Differ. Equations, 13 (2001), 689. [16] P. Lubowiecki and H. Żołądek, The Hess-Appelrot system. II. Perturbation and limit cycles,, J. Differential Equations, 252 (2012), 1701. doi: 10.1016/j.jde.2011.06.012. [17] A. J. Maciejewski and M. Przybylska, Differential Galois theory approach to the non-integrability of the heavy top,, Ann. Fac. Sci. Touluse, 14 (2005), 123. [18] N. A. Nekrasov, Analytic investigation of a particular case of motion of a heavy rigid body about fixed point,, Matem. Sbornik, 18 (1895), 162. [19] Yu. P. Varkhalev and G. V. Gorr, Asymptotically pendulum motions of the Hess-Appelrot gyroscope,, Prikl. Mat. Mekh., 48 (1984), 490. [20] N. E. Zhukovski, Geometrische interpretation des Hess'schen falles der bewegung eines schweren starren korpers um eine festen punkt,, Jahr. Berichte Deutschen Math. Verein., 3 (1894), 62. [21] S. L. Ziglin, Branching of solutions and non-existence of first integrals in Hamiltonian mechanics. I,, Funct. Anal. Appl., 16 (1983), 181. [22] H. Żołądek, "The Monodromy Group,", Birkhäuser, (2006).
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