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The phenomenon of reversal in the Euler-Poincaré-Suslov nonholonomic systems. (English) Zbl 1418.37108
Summary: The development of robotics makes it necessary to study the problem of controlling nonholonomic systems [M. Svinin et al., Regul. Chaotic Dyn. 18, No. 1–2, 126–143 (2013; Zbl 1272.70049); A. V. Borisov et al., Regul. Chaotic Dyn. 18, No. 1–2, 144–158 (2013; Zbl 1303.37021); T. B. Ivanova and E. N. Pivovarova, Regul. Chaotic Dyn. 19, No. 1, 140–143 (2014; Zbl 1353.70059)]. In this paper, the dynamics of nonholonomic systems on Lie groups with a left-invariant kinetic energy and left-invariant constraints are considered. Equations of motion form a closed system of differential equations on the corresponding Lie algebra. In addition, the effect of change in the stability of steady motions of these systems with the direction of motion reversed (the reversal found in rattleback dynamics) is discussed. As an illustration, the rotation of a rigid body with a fixed point and the Suslov nonholonomic constraint as well as the motion of the Chaplygin sleigh is considered.

MSC:
37J60 Nonholonomic dynamical systems
37J25 Stability problems for finite-dimensional Hamiltonian and Lagrangian systems
70E40 Integrable cases of motion in rigid body dynamics
70F25 Nonholonomic systems related to the dynamics of a system of particles
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
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[1] Svinin, M; Morinaga, A; Yamamoto M, On the dynamic model and motion planning for a spherical rolling robot actuated by orthogonal internal rotors, Regul Chaotic Dyn., 18, 126-143, (2013) · Zbl 1272.70049
[2] Borisov, AV; Kilin AA; Mamaev, IS, How to control the Chaplygin ball using rotors. II, Regul Chaotic Dyn., 18, 144-158, (2013) · Zbl 1303.37021
[3] Ivanova, TB; Pivovarova, EN, Comments on the paper by A. V. borisov, A. A. kilin, I. S. mamaev “how to control the Chaplygin ball using rotors. II”, Regul Chaotic Dyn., 19, 140-143, (2014) · Zbl 1353.70059
[4] Arnold, VI, Sur la géométrie différentielle des groups de Lie de dimension infinite et ses applications à l’hydrodynamique des fluides parfaits, Ann Inst Fourier, 16, 319-361, (1966) · Zbl 0148.45301
[5] Bolsinov, AV; Borisov, AV; Mamaev, IS, Hamiltonization of nonholonomic systems in the neighborhood of invariant manifolds, Regul Chaotic Dyn, 16, 443-464, (2011) · Zbl 1309.37049
[6] Borisov, AV; Fedorov, YN; Mamaev, IS, Chaplygin ball over a fixed sphere: an explicit integration, Regul Chaotic Dyn, 13, 557-571, (2008) · Zbl 1229.37080
[7] Borisov, AV; Jalnine, AY; Kuznetsov, SP; Sataev, IR; Sedova, JV, Dynamical phenomena occurring due to phase volume compression in nonholonomic model of the rattleback, Regul Chaotic Dyn, 17, 512-532, (2012) · Zbl 1263.74021
[8] Borisov, AV; Mamaev, IS, The rolling motion of a rigid body on a plane and a sphere: hierarchy of dynamics, Regul Chaotic Dyn, 7, 177-200, (2002) · Zbl 1058.70009
[9] Borisov, AV; Mamaev, IS, Rolling of a non-homogeneous ball over a sphere without slipping and twisting, Regul Chaotic Dyn, 12, 153-159, (2007) · Zbl 1229.37081
[10] Borisov, AV; Mamaev, IS, Conservation laws, hierarchy of dynamics and explicit integration of nonholonomic systems, Regul Chaotic Dyn, 13, 443-490, (2008) · Zbl 1229.70038
[11] Caratheodory, C; Der Schlitten, Z, Angew, Math Mech, 13, 71-76, (1933)
[12] Fedorov, YN; Maciejewski, AJ; Przybylska, M, Suslov problem: integrability, meromorphic and hypergeometric solutions, Nonlinearity, 22, 2231-2259, (2009) · Zbl 1170.70007
[13] Jovanović, B, Some multidimensional integrable cases of nonholonomic rigid body dynamics, Regul Chaotic Dyn, 8, 125-132, (2003) · Zbl 1023.70002
[14] Kozlov VV, Fedorov YN. Various aspects of \(n\)-Dimensional rigid body dynamics, in dynamical systems in classical mechanics. Amer Math Soc Transl Ser 2 1995;168:141-171. Providence, R.I.: AMS. · Zbl 0148.45301
[15] Poincaré, H; Sur une forme nouvelle des équations de la Mécanique, No article title, C R Acad Sci, 132, 369-371, (1901)
[16] Walker, GT, On a dynamical top, Quart J Pure Appl Math, 28, 174-185, (1896)
[17] Zenkov, DV; Bloch, AM, Dynamics of the n-dimensional Suslov problem, J Geom Phys, 34, 121-136, (2000) · Zbl 0989.70010
[18] Borisov AV, Kilin AA, Mamaev IS. Hamiltonian representation and integrability of the suslov problem. Nelin Dinam 2010;6(1):127-142. (Russian). · Zbl 1058.70009
[19] Borisov AV, Mamaev IS. Strange attractors in rattleback dynamics. Phys Uspekhi 2003;46(4):393-403. see also: Uspekhi Fiz. Nauk, 2003, 173, 4, 407-418. · Zbl 1229.37081
[20] Borisov AV, Mamaev IS. The dynamics of a chaplygin sleigh. J Appl Math Mech 2009;73(2):156-161. see also: Prikl. Mat. Mekh., 2009, 73, 2, 219-225.
[21] Kozlov VV. Realization of nonintegrable constraints in classical mechanics. Sov Phys Dokl 1983;28:735-737. see also: Dokl. Akad. Nauk SSSR, 1983, 272, 3, 550-554. · Zbl 0579.70014
[22] Kozlov VV. On the theory of integration of the equations of nonholonomic mechanics. Uspekhi Mekh 1985;8(3):85-107. (Russian). · Zbl 1170.70007
[23] Kozlov VV. Invariant measures of Euler-Poincaré equations on lie algebras. Funct Anal Appl 1988;22(1):58-59. see also: Funktsional. Anal. i Prilozhen., 1988, 22, 1, 69-70. · Zbl 1353.70059
[24] Kozlov, VV, The Euler-Jacobi-Lie integrability theorem, Regul Chaotic Dyn, 18, 329-343, (2013) · Zbl 1283.34035
[25] Lyapunov AM. 1950. The general problem of the stability of motion, Moscow: Gostekhizdat. (Russian).
[26] Markeev AP. 2014. Dynamics of a Body Touching a Rigid Surface, Moscow-Izhevsk: R&C Dynamics, Institute of Computer Science. (Russian). · Zbl 1272.70049
[27] Neimark JI, Fufaev NA. Dynamics of nonholonomic systems 1972;33. Providence, RI: AMS. · Zbl 1263.74021
[28] Suslov GK, Theoretical Mechanics Moscow: Gostekhizdat. 1946. Russian). · Zbl 1170.70007
[29] Chaplygin SA. On the theory of motion of nonholonomic systems. The reducing-multiplier theorem. Regul Chaotic Dyn 2008;13(4):369-376. see also: Mat. Sb., 1912, 28, 2, 303-314. · Zbl 1229.37082
[30] Fedorov, YN; Maciejewski, AJ; Przybylska, M, The Poisson equations in the nonholonomic Suslov problem: integrability, meromorphic and hypergeometric solutions, Nonlinearity, 22, 2231-2259, (2009) · Zbl 1170.70007
[31] Borisov, AV; Kazakov, AO; Sataev, IR, The reversal and chaotic attractor in the nonholonomic model of chaplygins top, Regul Chaotic Dyn, 19, 718-733, (2014) · Zbl 1358.70006
[32] Gonchenko, AS; Gonchenko, SV; Kazakov, AO, Richness of chaotic dynamics in nonholonomic models of a celtic stone, Regul Chaotic Dyn, 18, 521-538, (2013) · Zbl 1417.37222
[33] Borisov, AV; Mamaev, IS; Bizyaev, IA, The hierarchy of dynamics of a rigid body rolling without slipping and spinning on a plane and a sphere, Regul Chaotic Dyn, 18, 277-328, (2013) · Zbl 1367.70007
[34] Borisov, AV; Mamaev, IS; Bizyaev, IA, The Jacobi integral in nonholonomic mechanics, Regul Chaotic Dyn, 20, 383-400, (2015) · Zbl 1367.70036
[35] Bizyaev, IA; Borisov, AV; Mamaev, IS, The dynamics of nonholonomic systems consisting of a spherical shell with a moving rigid body inside, Regul Chaotic Dyn, 19, 198-213, (2014) · Zbl 1308.70003
[36] Kazakov, AO, Strange attractors and mixed dynamics in the problem of an unbalanced rubber ball rolling on a plane, Regul Chaotic Dyn., 18, 508-520, (2013) · Zbl 1417.37223
[37] Kozlov, VV, Remarks on integrable systems, Regul Chaotic Dyn., 19, 145-161, (2014) · Zbl 1382.34002
[38] Borisov, AV; Kilin, AA; Mamaev, IS, The problem of drift and recurrence for the rolling Chaplygin ball, Regul Chaotic Dyn., 18, 832-859, (2013) · Zbl 1286.70007
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