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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.

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
Full Text: DOI
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