Alexander Rodnikov

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.

We study a particle equilibria with respect to axes of precession and of dynamical symmetry of a rigid body in assumption that the body gravitational field is composed of gravitational fields of two conjugate complex masses being on imaginary distance. We establish that there are not more then two of these equilibria in the plane passing the body mass center orthogonally to the precession axis. Using terminology of the Generalized Restricted Circular Problem of Three Bodies, we call these equilibria the Triangular Libration Points (TLP).We find TLPs’ coordinates analytically and we trace their evolution at changing values of the system parameters. We also prove that TLPs are instable.

A particle relative equilibria near a rigid body in the plane passing through the body axes of precession and of dynamical symmetry are studied in assumption that the body gravitational field can be composed as gravitational field of two conjugate complex masses being on imaginary distance. Using terminology of the Generalized Restricted Circular Problem of Three Bodies, these equilibria are called Coplanar Libration Points (CLP). One can show that CLP set is divided into three subsets dependently on CLPs type of evolution. There are 2 «external» CLPs going from infinity to the rigid body if precession angular velocity goes from zero to infinity, from 2 to 6 «internal» CLPs between axis of precession and axis of dynamical symmetry, and from 0 to 3 «central» CLPs near singular points of gravitational potential. Numerical-analitical algorithm of CLPs coordinates computation is suggested. This algorithm is based on some special trigonometrical transformations of coordinates and parameters.

A particle steady motions in vicinity of dynamically symmetric precessing rigid body are studied in assumption that the body gravitational field is modeled as gravitational field of two centers being on imaginary distance. Such particle motion equations are a variant of motion equations of the Generalized Restricted Circular Problem of Three Bodies (GRCP3B). The number of Coplanar Libration Points, i.e. the particle equilibria in the plane passing through the body axis of dynamical symmetry and through the axis of precession are established. (This number is odd and can be equal to 5, 7 or 9). CLPs evolution are studied at changing values of the considered system parameters. Moreover, two Triangular Libration Points, i. e. the particle equilibria in the axis crossing the body mass center orthogonally to axes of precession and dynamical symmetry are found.

We study a space station equilibria on the cable called «the leier» with ends placed in poles of a dynamically-symmetric asteroid. We suggest some criteria of these equilibria stability for the station fixed on the leier. Using condition for V.V. Beletsky’s Generalized Restricted Circular Problem of Three Bodies we classify coplanar equilibria, i.e. equilibria in the plane composed by axes of dynamical symmetry and precession if the asteroid gravitational field is close to gravitational field of two particles of equal masses.

A particle steady motions in vicinity of dynamically symmetric precessing rigid body are studied in assumption that the body gravitational field is modeled as two centers gravitational field. The particle motion equations are written as two-parametric generalization for equations of Restricted Circular Problem of Three Bodies (RCP3B). Existence and number of the particle relative equilibria in the plane passing through the body axis of dynamical symmetry and through the vector of angular momentum are established. These equilibria called Coplanar Libration Points (CLP) are analogs of Eulerian Libration Points in RCP3B. Stability of CLP is studied for the first approximation in assumption that attracting centers have equal masses.

We study a particle motion along a cable with ends fixed in a precessed rigid body. Such cable called «the leier» is a model of space elevator for a dynamically symmetric asteroid. (The Dutch term «leier» means the rope with fixed ends). In this paper we find two integrable cases of the particle motion equations (for zero and right nutation angle) Phase portraits for integrable situations are built taking into account conditions of motion with the tense cable and assuming the body gravitation is close to gravitational field of two equal point masses that are in the axis of dynamical symmetry. Using «the Generalized Restricted Circular Problem of Three Bodies» by V. V. Beletsky, we study the particle equilibria on the leier in the plane containing the body mass center and being perpendicular to the precession axis for all possible nutation angles. Some facts on these equilibria stability are formulated.

We consider the system moving in the Newtonian central force field and consisting of a dumbbell satellite and a particle. The particle coasts along on the cable with ends placed in the dumbbell endpoints. We call such cable a «leier». We suppose the system mass center describes circular orbit, the particle mass is small in comparison with the dumbbell mass and the cable length is small in comparison with orbit radius. Assuming the cable don’t leave the orbit plane we study the dumbbell rotations forced by the particle. We note that the particle sufficiently influence the dumbbell motion only in vicinity of the dumbbell rotation separatrix. We claim that there exist a set of the dumbbell unstable asymptotic motions tending to librations about the orbit tangent. Initial conditions for these motions compose a surface in the system phase space. We deduce an equation approximating this surface. We consider this equation as a criterion for the direction of the dumbbell rotation from the vicinity of unstable equilibria. We deduce formulae approximating the dumbbell near-separatrix motion if the cable is rather long and the dumbbell is composed of equal masses. Using numerical procedures, we analyse the dumbbell motion in two-dimensional transections of four-dimensional space of initial conditions if the cable is rather short.