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In this work, we aim to introduce and investigate a novel type of ER3BP with variable angular momentum of secondary planet correlated with the motion of test particle in the pulsating synodic frame, which will maintain its orbit located in the proximity of secondary body. Indeed, tidal phenomena govern the dynamics of slow change of the eccentricity of the secondary planet orbit related to the fact that they have been changing its rotational angular momentum during a long-time period of secondary planet’s motion around the primary on quasi-stable elliptic orbit. Both novel theoretical and numerical findings are presented as well.

In this paper, we present a new approach for solving equations of motion for the trapped motion of the infinitesimal mass m in case of the elliptic restricted problem of three bodies (ER3BP) (primaries $$M_\mathrm{Sun}$$ and $$m_\mathrm{planet}$$ are rotating around their common centre of masses on elliptic orbit): a new type of the solving procedure is implemented here for solving equations of motion of the infinitesimal mass m in the vicinity of the barycenter of masses $$M_\mathrm{Sun}$$ and $$m_\mathrm{planet}$$ . Meanwhile, the system of equations of motion has been successfully explored with respect to the existence of analytical way for presentation of the approximated solution. As the main result, equations of motion are reduced to the system of three nonlinear ordinary differential equations: (1) equation for coordinate x is proved to be a kind of appropriate equation for the forced oscillations during a long-time period of quasi-oscillations (with a proper restriction to the mass $$m_\mathrm{planet}$$ ), (2) equation for coordinate y reveals that motion is not stable with respect to this coordinate and condition $$y \sim 0$$ would be valid if only we choose zero initial conditions, and (3) equation for coordinate z is proved to be Riccati ODE of the first kind. Thus, infinitesimal mass m should escape from vicinity of common centre of masses $$M_\mathrm{Sun}$$ and $$m_\mathrm{planet}$$ as soon as the true anomaly f increases insofar. The main aim of the current research is to point out a clear formulation of solving algorithm or semi-analytical procedure with partial cases of solutions to the system of equations under consideration. Here, semi-analytical solution should be treated as numerical algorithm for a system of ordinary differential equations (ER3BP) with well-known code for solving to be presented in the final form.

In this paper, we present a new ansatz for solving equations of motion for the trapped orbits of the infinitesimal mass m, which is moving near the primary M3 in case of bi-elliptic restricted problem of four bodies (BiER4BP), where three primaries M1, M2, M3 are rotating around their common center of mass on elliptic orbits with hierarchical configuration M3 ≪ M2 ≪ M1. A new type of the solving procedure is implemented here to obtain the coordinates $$ \vec{r} = \;\{ x,y,z\} $$ of the infinitesimal mass m with its orbit located near the primary M3. Meanwhile, the system of equations of motion has been successfully explored with respect to the existence of analytical or semi-analytical (approximated) way for presentation of the solution. We obtain as follows: (1) the solution for coordinate x is described by the key nonlinear ordinary differential equation of fourth order at simplifying assumptions, (2) solution for coordinate y is given by the proper analytical expression, depending on coordinate x and true anomaly f, (3) the expression for coordinate z is given by the equation of Riccati-type—it means that coordinate z should be quasi-periodically oscillating close to the fixed plane $$ \{ x,y,\,0\} $$ .

The time-optimal deceleration of a dynamically asymmetric body is studied using nondimensional variables, which makes it possible to obtain a multi-parameter system of equations of motion. A vector hodograph of the angular momentum is modeled in a three-dimensional space for different values of the system parameters. It is concluded that certain ratios between the problem parameters are needed for the optimal deceleration of the body.

We study the fast rotational motion of a dynamically asymmetric satellite with a spherical cavity filled with a highly viscous liquid about the center of mass under the action of gravitational torque and medium drag torques. The system obtained by averaging over the Euler–Poinsotmotion and by using a modified averaging method is analyzed. An analytic study and numerical analysis are carried out.

Rotational motion around the center of mass of a dynamically asymmetric satellite with spherical cavity filled with viscous liquid is studied for low Reynolds numbers. The numeric analysis of the vector change of kinetic momentum of a solid body was performed, the hodograph of the vector was plotted, and numerical study of the stability of the extreme position of the Eigen rotation axis of a solid body was carried out. A solid body with mass geometry of Earth was studied.

A minimum-time problem on deceleration of rotations of a free rigid body affected by a small control torque with close but not identical coefficients is studied; such a problem can be considered as a quasi-optimal control problem. In addition, the rigid body is affected by a small deceleration viscous friction torque. The body is assumed to be dynamically asymmetric. A quasi-optimal feedback control for the deceleration of rotations of the rigid body is constructed, the optimal control time, and phase trajectories are found. The quasi-stationary trajectories are analyzed.

A minimum-time problem on deceleration of rotations of a free rigid body is studied. It is assumed that the body contains a spherical cavity filled with highly viscous fluid. The body is subjected to a retarding torque of viscous friction. It is assumed that the body is dynamically asymmetric. An optimal control law for the deceleration of rotations of the body is synthesized, and the corresponding time and phase trajectories are determined.

Rapid rotational motion of a dynamically asymmetric satellite relative to the center of mass is studied. The satellite has a cavity filled with viscous fluid at low Reynolds numbers, and it moves under the action of moments of gravity and light pressure forces. Orbital motions with an arbitrary eccentricity are supposed to be specified. The system, obtained after averaging over the Euler-Poinsot motion and applying the modified averaging method, is analyzed. The numerical analysis in the general case is performed, and the analytical study in the axial rotation vicinity is carried out. The motion in the specific case of a dynamically symmetric satellite is considered.

A minimum-time problem on deceleration of rotation of a free rigid body is studied. The body is assumed to contain a viscous-elastic element, which is modeled as a movable point mass attached to the body via a damper. In addition, the body is subjected to a retarding torque generated by linear medium resistance forces. In an undeformed state, the body is assumed to be dynamically symmetric, with the mass being located on the symmetry axis. An optimal control law for deceleration of rotation of the body is synthesized, and the corresponding time and phase trajectories are determined.

The problem of time-optimal deceleration of rotation of a free solid body is studied. It is assumed that the body contains a spherical cavity filled with highly viscous liquid. Low decelerating moment of viscous friction forces also acts on the solid body. It is assumed that the body is dynamically symmetric. The optimal control law for deceleration of rotation of the carrier solid body in the form of synthesis, the operation time, and the phase trajectories are determined.

We study the fast rotational motion of a dynamically nonsymmetric satellite about the center of mass under the action of the gravitational torque and the drag torque. Orbital motions with arbitrary eccentricity are assumed to be given. The drag torque is assumed to be a linear function of the angular velocity. The system obtained after the averaging over the Euler-Poinsot motion is studied. We discover the following phenomena: the modulus of the angular momentum and the kinetic energy decrease, and there exist quasistationary regimes of motion (along the polhodes). The orientation of the angular momentum vector in the orbital frame of reference is determined. The general case is studied numerically, and an analytic study is performed in a neighborhood of the axial rotation and in the case of small dissipation.