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The dynamics of systems with servoconstraints. I. (English) Zbl 1353.70036
Regul. Chaotic Dyn. 20, No. 3, 205-224 (2015); translation in Nelineĭn. Din. 11, No. 2, 353-376 (2015).
Summary: The paper discusses the dynamics of systems with Béghin’s servoconstraints where the constraints are realized by means of controlled forces. Classical nonholonomic systems are an important particular case. Special attention is given to the study of motion on Lie groups with left-invariant kinetic energy and left-invariant constraints. The presence of symmetries allows one to reduce the dynamic equations to a closed system of differential equations with quadratic right-hand sides on a Lie algebra. Examples are given which include the rotation of a rigid body with a left-invariant servoconstraint – the projection of the angular velocity onto some direction fixed in the body is equal to zero (a generalization of the nonholonomic Suslov problem) – and the motion of the Chaplygin sleigh with servoconstraints of a certain type. The dynamics of systems with Béghin’s servoconstraints is richer and more varied than the more usual dynamics of nonholonomic systems.
Part II, see Regul. Chaotic Dyn. 20, No. 4, 401–427 (2015); translation in Nelineĭn. Din. 11, No. 3, 579–611 (2015; Zbl 1353.70012).

MSC:
70F25 Nonholonomic systems related to the dynamics of a system of particles
70Q05 Control of mechanical systems
34D20 Stability of solutions to ordinary differential equations
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