Boris Bardin
Volokolamskoe sh. 4, Moscow, 125993, Russia
Moscow Aviation Institute (National Research University)
Professor of Moscow Aviation Institute (National Research University), Doctor of Physics and Mathematics
Chief of Department of Theoretical Physics, superviser of speciality "Application of mathematical methods in the problems of aerospace engineering" at MAI
Publications:
Bardin B. S., Sukhov E. A., Volkov E. V.
Nonlinear Orbital Stability of Periodic Motions in the Planar Restricted FourBody Problem
2023, Vol. 19, no. 4, pp. 545557
Abstract
We consider the planar circular restricted fourbody problem with a small body of negligible
mass moving in the Newtonian gravitational field of three primary bodies, which form a stable
Lagrangian triangle. The small body moves in the same plane with the primaries. We assume
that two of the primaries have equal masses. In this case the small body has three relative
equilibrium positions located on the central bisector of the Lagrangian triangle. In this work we study the nonlinear orbital stability problem for periodic motions emanating from the stable relative equilibrium. To describe motions of the small body in a neighborhood of its periodic orbit, we introduce the socalled local variables. Then we reduce the orbital stability problem to the stability problem of a stationary point of symplectic mapping generated by the system phase flow on the energy level corresponding to the unperturbed periodic motion. This allows rigorous conclusions to be drawn on orbital stability for both the nonresonant and the resonant cases. We apply this method to investigate orbital stability in the case of third and fourthorder resonances as well as in the nonresonant case. The results of the study are presented in the form of a stability diagram. 
Bardin B. S., Chekina E. A., Chekin A. M.
On the Orbital Stability of Pendulum Oscillations of a Dynamically Symmetric Satellite
2022, Vol. 18, no. 4, pp. 589607
Abstract
The orbital stability of planar pendulumlike oscillations of a satellite about its center of
mass is investigated. The satellite is supposed to be a dynamically symmetrical rigid body
whose center of mass moves in a circular orbit. Using the recently developed approach [1], local
variables are introduced and equations of perturbed motion are obtained in a Hamiltonian form.
On the basis of the method of normal forms and KAM theory, a nonlinear analysis is performed
and rigorous conclusions on orbital stability are obtained for almost all parameter values. In
particular, the socalled case of degeneracy, when it is necessary to take into account terms of
order six in the expansion of the Hamiltonian function, is studied.

Bardin B. S., Avdyushkin A. N.
On Stability of the Collinear Libration Point $L_1^{}$ in the Planar Restricted Circular Photogravitational ThreeBody Problem
2022, Vol. 18, no. 4, pp. 543562
Abstract
The stability of the collinear libration point $L_1^{}$ in the photogravitational threebody problem
is investigated. This problem is concerned with the motion of a body of infinitely small mass
which experiences gravitational forces and repulsive forces of radiation pressure coming from two
massive bodies. It is assumed that the massive bodies move in circular orbits and that the body
of small mass is located in the plane of their motion. Using methods of normal forms and KAM
theory, a rigorous analysis of the Lyapunov stability of the collinear libration point lying on the
segment connecting the massive bodies is performed. Conclusions on the stability are drawn
both for the nonresonant case and for the case of resonances through order four.

Bardin B. S., Chekina E. A.
On the Orbital Stability of Pendulumlike Oscillations of a Heavy Rigid Body with a Fixed Point in the Bobylev – Steklov Case
2021, Vol. 17, no. 4, pp. 453464
Abstract
The orbital stability of pendulumlike oscillations of a heavy rigid body with a fixed point in
the Bobylev – Steklov case is investigated. In particular, a nonlinear study of the orbital stability
is performed for the socalled case of degeneracy, where it is necessary to take into account terms
of order six in the Hamiltonian expansion in a neighborhood of the unperturbed periodic orbit.

Bardin B. S.
On a Method of Introducing Local Coordinates in the Problem of the Orbital Stability of Planar Periodic Motions of a Rigid Body
2020, Vol. 16, no. 4, pp. 581594
Abstract
A method is presented of constructing a nonlinear canonical change of variables which
makes it possible to introduce local coordinates in a neighborhood of periodic motions of an
autonomous Hamiltonian system with two degrees of freedom. The problem of the orbital stability
of pendulumlike oscillations of a heavy rigid body with a fixed point in the Bobylev – Steklov
case is discussed as an application. The nonlinear analysis of orbital stability is carried out
including terms through degree six in the expansion of the Hamiltonian function in a neighborhood
of the unperturbed periodic motion. This makes it possible to draw rigorous conclusions
on orbital stability for the parameter values corresponding to degeneracy of terms of degree four
in the normal form of the Hamiltonian function of equations of perturbed motion.

Bardin B. S., Chekina E. A.
On Orbital Stability of Pendulumlike Satellite Rotations at the Boundaries of Stability Regions
2019, Vol. 15, no. 4, pp. 415428
Abstract
The motion of a rigid body satellite about its center of mass is considered. The problem
of the orbital stability of planar pendulumlike rotations of the satellite is investigated. It is
assumed that the satellite moves in a circular orbit and its geometry of mass corresponds to
a plate. In unperturbed motion the minor axis of the inertia ellipsoid lies in the orbital plane. A nonlinear analysis of the orbital stability for previously unexplored values of parameters corresponding to the boundaries of the stability regions is carried out. The study is based on the normal form technique. In the special case of fast rotations a normalization procedure is performed analytically. In the general case the coefficients of normal form are calculated numerically. It is shown that in the case under consideration the planar rotations of the satellite are mainly unstable, and only on one of the boundary curves there is a segment where the formal orbital stability takes place. 
Bardin B. S., Panev A. S.
On the Motion of a Body with a Moving Internal Mass on a Rough Horizontal Plane
2018, Vol. 14, no. 4, pp. 519542
Abstract
We consider a vibrationdriven system which consists of a rigid body and an internal mass.
The internal mass is a particle moving in a circle inside the body. The center of the circle is
located at the mass center of the body and the absolute value of particle velocity is a constant.
The body performs rectilinear motion on a horizontal plane, whereas the particle moves in
a vertical plane. We suppose that dry friction acts between the plane and the body. We have investigated the dynamics of the above system in detail and given a full description of the body’s motion for any values of its initial velocity. In particular, it is shown that there always exists a periodic mode of motion. Depending on parameter values, one of three types of this periodic mode takes place. At any initial velocity the body either enters a periodic mode during a finite time interval or it asymptotically approaches the periodic mode. 
Bardin B. S., Chekina E. A.
On the stability of planar oscillations of a satelliteplate in the case of essential type resonance
2017, Vol. 13, No. 4, pp. 465–476
Abstract
We consider satellite motion about its center of mass in a circle orbit. We study the problem of orbital stability for planar pendulumlike oscillations of the satellite. It is supposed that the satellite is a rigid body whose mass geometry is that of a plate. We assume that on the unperturbed motion the middle or minor inertia axis of the satellite lies in the orbit plane, i.e., the plane of the satelliteplate is perpendicular to the plane of the orbit. In this paper we perform a nonlinear analysis of the orbital stability of planar pendulumlike oscillations of a satelliteplate for previously unexplored parameter values corresponding to the boundaries of regions of stability in the first approximation, where the essential type resonances take place. It is proved that on the mentioned boundaries the planar pendulumlike oscillations are formally orbital stable or orbitally stable in third approximation. 
Bardin B. S., Chekina E. A.
On the stability of a resonant rotation of a satellite in an elliptic orbit
2016, Vol. 12, No. 4, pp. 619–632
Abstract
We deal with the problem of stability for a resonant rotation of a satellite. It is supposed that the satellite is a rigid body whose center of mass moves in an elliptic orbit. The resonant rotation is a planar motion such that the body completes one rotation in absolute space during two orbital revolutions of its center of mass. The stability analysis of the resonant rotation with respect to planar perturbations has been performed in detail earlier. In this paper we investigate the stability of the resonant rotation with respect to both planar and spatial perturbations for a nonsymmetric satellite. For small values of the eccentricity we have obtained boundaries of instability domains (parametric resonance domains) in an analytic form. For arbitrary eccentricity values we numerically construct domains of stability in linear approximation. Outside the above stability domains the resonant rotation is unstable in the sense of Lyapunov.

Bardin B. S., Savin A. A.
On orbital stability pendulumlike oscillations and rotation of symmetric rigid body with a fixed point
2012, Vol. 8, No. 2, pp. 249266
Abstract
We deal with the problem of orbital stability of planar periodic motions of a heavy rigid body with a fixed point. We suppose that the mass center of the body is located in the equatorial plane of the inertia ellipsoid. Unperturbed motions represent oscillations or rotations of the body around a principal axis, keeping a fixed horizontal position.
Local coordinates are introduced in a neighborhood of the unperturbed periodic motion and equations of perturbed motion are obtained in Hamiltonian form. Domains of orbital instability are established by means of linear analysis. Outside of the above domains nonlinear study is performed. The nonlinear stability problem is reduced to a stability problem of a fixed point of symplectic map generated by the equations of perturbed motion. Coefficients of the above map are obtained numerically. By analyzing of the coefficients mentioned rigorous results on orbital stability or instability are obtained. In the case of oscillations with small amplitudes as well as in the case of rotations with high angular velocities the problem of orbital stability is studied analytically. 
Bardin B. S.
On orbital stability of pendulum like motions of a rigid body in the BobylevSteklov case
2009, Vol. 5, No. 4, pp. 535550
Abstract
We deal with the problem of orbital stability of pendulum like periodic motions of a heavy rigid body with a fixed point. We suppose that the geometry of the mass of the body corresponds to the Bobylev—Steklov case. Unperturbed motion represents oscillations or rotations of the body around a principal axis, occupying a fixed horizontal position. The problem of the orbital stability is considered on the base of a nonlinear analysis. In the case of oscillations with small amplitudes as well as in the case of rotations with high angular velocities we studied the problem analytically. In general case we reduce the problem to the stability study of fixed point of the symplectic map generated by equations of perturbed motion. We calculate coefficients of the symplectic map numerically. By analyzing of the coefficients mentioned we establish orbital stability or instability of the unperturbed motion. The results of the study are represented in the form of stability diagram. 
Bardin B. S.
On nonlinear oscillations of Hamiltonian system in case of fourth order resonance
2007, Vol. 3, No. 1, pp. 5774
Abstract
We deal with an autonomous Hamiltonian system with two degrees of freedom. We assume that the Hamiltonian function is analytic in a neighborhood of the phase space origin which is an equilibrium point. We consider the case when two imaginary eigenvalues of the matrix of the linearized system are in the ratio 3:1. We study nonlinear conditionallyperiodic motions of the system in the vicinity of the equilibrium point. Omitting the terms of order higher then five in the normalized Hamiltonian we analyze the socalled truncated system in detail. We show that its general solution can be given in terms of elliptic integrals and elliptic functions. The motions of truncated system are either periodic, or asymptotic to a periodic one, or conditionallyperiodic. By using the KAM theory methods we show that the most of conditionallyperiodic trajectories of the truncated systems persist also in the full system. Moreover, the trajectories that became not conditionallyperiodic in the full system belong to a subset of exponentially small measure. The results of the study are applied for the analysis of nonlinear motions of a symmetric satellite in a neighborhood of its cylindric precession.
