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On spacial motions of an orbital tethered system. (Russian. English summary) Zbl 1382.37093
Summary: We study motions of a particle along a rope with ends fixed to an extended rigid body whose center of mass traces out a circular orbit in the central Newtonian force field. (Such a rope is called a tether.) We assume that the tether realizes an ideal unilateral constraint. We derive particle motion equations on the surface of the ellipsoid, which restricts the particle motion, and conditions that guarantee such motions. We also study the existence and stability of relative equilibria of the particle with respect to the orbital frame of reference. We prove stability of the integral manifold of the particle motions in the plane of the orbit. We note that small-amplitude librations near this manifold can be described by approximate equations that can be reduced to Riccati’s equation. We establish that generally the spacial motions of the particle are chaotic for initial conditions from some vicinity of the separatrix motion in the plane of the orbit and are regular in other cases. We also note that chaotic motions usually lead to a situation where the particle comes off the constraint, in other words, to motions inside the above-mentioned ellipsoid.

##### MSC:
 37N05 Dynamical systems in classical and celestial mechanics 70F15 Celestial mechanics 70F20 Holonomic systems related to the dynamics of a system of particles 70K55 Transition to stochasticity (chaotic behavior) for nonlinear problems in mechanics
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