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Perturbed rotational rigid body motions similar to the case of Lagrange top subjected to restoring and perturbation slowly time-varying torques of forces are studied in this paper. The restoring torque also depends on the small angle of nutation. The following problem is formulated in the current research: analyzing the solution’s behavior of system of motion equations for the values of small parameter different from zero on a sufficiently large interval of time. An averaging method is used for solving the problem. We have established the terms for feasibility to average (with respect to phase of nutation angle) the equations of rigid body motion related to Lagrange case. The averaged system of equations is received in the first approximation. Asymptotic approach permits to obtain some qualitative results and to describe evolution of motion using simplified averaged equations. In the case of rotational motion of a body immersed in the linear-dissipative medium, the numerical integration of the averaged system of equations is conducted. A new class of rotational motions of the Lagrange top is studied for an unsteady perturbation torque, as well as restoring torque that slowly varies with time and depends on the small nutation angle. These results are an extension of our former works (in the absence of slow time τ in the expressions for μ and M i or μ, where M i are dependent on τ only).

We have presented in this analytical research the revisiting of approach for mathematical modeling the Glacier dynamics in terms of viscous-plastic theory of 2-dimensional movements within (x, y)-plane in cartesian coordinates. The stationary creeping approximation for the plane-parallel flow of slowly moving glacial ice on absolutely flat surface without any inclination has been considered. Even in such simple formulation, equations of motion that governs by the dynamics of viscous-plastic flow of glacial ice is hard to be solved analytically. We have succeeded in obtaining analytical expression for the components of velocity in Ox-direction of motion for slowly moving glacial ice (Ox-axis coincides to the initial main direction of slowly moving glacial ice). Restrictions on the form of flow stem from the continuity equation as well as from the special condition for non-Newtonian (viscous-plastic) flow have been used insofar.

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.

We have presented in this communication a new solving procedure for Kelvin-Kirchhoff equations, considering the dynamics of the falling or ascending the rigid disc in an ideal incompressible fluid, assuming additionally the dynamical symmetry of rotation for the falling or ascending body, I_1 = I_2. Fundamental law of angular momentum conservation has been used for the aforementioned solving procedure. The system of Euler equations for dynamics of disc rotation has been explored in regard to the existence of an analytic way of presentation for the approximated solution (where we consider the case of laminar flow at slow regime of deceleration of the symmetric disc rotation). This scenario is associated with consideration of that the Stokes boundary layer phenomenon on the boundaries of the body has also been assumed at formulation of basic Kelvin-Kirchhoff equations. It allows us to use the results of Jeffery fundamental work, in which analytical solution for rotations of spheroid particle in Euler angles was obtained. The results of calculations for the components of angular velocity should then be used for solving momentum equation of Kelvin-Kirchhoff system.

We have presented in the current analytic research the generating formulae and results of direct mathematical modelling of non-classical trends for COVID-19’s evolution in world which, nevertheless, can be divided into two types: (1) the general trends for European countries such as Germany presented by the curve of modified sigmoid-type with up-inclination of the upper limit of saturation (at the end of first wave of pandemic) as well as for other cases of key countries that suffered from pandemic such as USA, India, Brazil, Russia (we conclude that the same type of coronavirus pandemic is valid for most of the countries in world with similar scenarios of the same type for general trends); (2) non-classical general trends for Middle East countries (such as Iran), with the appropriate bulge on graphical plots at the beginning of first wave of pandemic. We expect that the second wave of pandemic will pass its peak at the end of December 2020 for various countries. Moreover, the second wave of pandemic will have come to end at first decade of January 2021 in Germany and Iran (but at the end of January 2021 in India as well), so we should restrict ourselves in modelling the first and second waves of pandemic within this time period for these countries. Thus, the model of first approximation is considered here which allows to understand the mean-time trends of COVID-19 evolution for the first + second waves of pandemic for USA, Brazil and Russia, or predict the approximated time period of the upcoming third wave of pandemic in cases of India, Germany and Iran.

The paper develops an approximate solution to the system of Euler’s equations with additional perturbation term for dynamically symmetric rotating rigid body. The perturbed motions of a rigid body, close to Lagrange’s case, under the action of restoring and perturbation torques that are slowly varying in time are investigated. We describe an averaging procedure for slow variables of a rigid body perturbed motion, similar to Lagrange top. Conditions for the possibility of averaging the equations of motion with respect to the nutation phase angle are presented. The averaging technique reduces the system order from 6 to 3 and does not contain fast oscillations. An example of motion of the body using linearly dissipative torques is worked out to demonstrate the use of general equations. The numerical integration of the averaged system of equations is conducted of the body motion. The graphical presentations of the solutions are represented and discussed. A new class of rotations of a dynamically symmetric rigid body about a fixed point with account for a nonstationary perturbation torque, as well as for a restoring torque that slowly varies with time, is studied. The main objective of this paper is to extend the previous results for problem of the dynamic motion of a symmetric rigid body subjected to perturbation and restoring torques. The proposed averaging method is implemented to receive the averaging system of equations of motion. The graphical representations of the solutions are presented and discussed. The attained results are a generalization of our former works where µ and Mi are independent of the slow time τ and Mi depend on the slow time only.

In this paper, we present a new ansatz for solving equations of motion for the trapped orbits of the infinitesimal mass (satellite), which is locked in the space trap to be moving near the planet in case of the elliptic restricted problem of three bodies, ER3BP (with Keplerian elliptic trajectories of primaries Sun and planet around each other). A new type of the solving procedure is implemented here to obtain the coordinates of the infinitesimal mass (satellite) with its orbit located near the planet. The system of equations of motion was applied for obtaining of the semi-analytic and analytic solutions. It is obtained that two cartesian coordinates (in a plane of mutual rotation of primaries Sun and planet around each other) depend on the true anomaly and a function which determines the quasi periodic character of solution, while the third coordinate (perpendicular to the plane of rotation of primaries) is quasi-periodically varying with true anomaly., Comment: 35 pages, 4 Tables, 5 Figures; Key words: elliptic restricted three-body problem (ER3BP), trapped motion, forced oscillations; Cite this article as: Ershkov, S., Rachinskaya, A. Arch. Appl. Mech. (2020). https://doi.org/10.1007/s00419-020-01829-6

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.

We have presented in this communication a new solving procedure for Kelvin–Kirchhoff equations, considering the dynamics of rising the quasi-rigid ellipsoid of rotation in an ideal incompressible fluid (up to the surface), assuming additionally the dynamical symmetry of rotation for the rising body, I 1 = I 2 . Fundamental law of angular momentum conservation has been used for the aforementioned solving procedure. The system of Euler equations for dynamics of non-rigid ellipsoid rotation has been explored in regard to the existence of an analytic way of presentation for the approximated solution (where we suppose that components of fluid’s torque vector { T i } are approximately proportional to the appropriate components of angular velocity { Ω i } ). The results of calculations for the components of angular velocity { Ω i } should then be used for solving momentum equation of Kelvin–Kirchhoff system. Thus, the full system of equations of Kelvin–Kirchhoff problem has been explored with respect to the existence of an analytic way of presentation of the general solution. The last but not least, we have pointed out the 1-st integral of Kelvin–Kirchhoff system under the aforesaid additional assumption for fluid’s torque vector { T i } when { I i } = const (but without the additional restriction of dynamical symmetry onto the form of the rising body, I 1 = I 2 ).

We have presented here our reply on the comments on our recent article “ON THE DYNAMICS OF NON-RIGID ASTEROID ROTATION”. We would restate again: our partial solution stems from fundamental law of angular momentum conservation for the non-rigid asteroid rotation (where asteroids were considered preferably as deformed ‘rubble pile’ asteroids, not as elastic bodies), meanwhile authors of the aforementioned comments have considered other partial solving procedure which can be associated with commensurable elongations of the asteroid during the deformations. This assumption contradicts the physical nature of most known asteroids (‘rubble pile’ asteroids are known to have a non-elastic internal rheology), therefore as such the solutions of Euler equations for dynamics of non-rigid asteroid rotation, obtained by authors of the comments, does not have real meaning in astrophysical sense. Despite the fact that it is useless in astrophysical area, the suggested strategy or solving procedure for the constructing a new class of special kind of solutions for Euler equations (dynamics of non-rigid elastic body's rotation) can be of partial interest to a journals for theoretical mechanics but not first-class astrophysical journals or journal in the field of celestial mechanics. So, we wish the best for the authors to possible publishing their full solution (not a simple a note, which seems to be “following-in-wake” in regard to our article) where they could discuss additionally the stability of solutions if it is deemed required.

In fluid mechanics, a lot of authors have been executing their researches to obtain the analytical solutions of Navier-Stokes equations. But there is an essential deficiency of non-stationary solutions indeed.In our presentation, we proceed exploring the case of non-stationary helical flows of the Navier-Stokes equations for incompressible fluids, with variable (spatially dependent) coefficient of proportionality α between velocity and the curl field of flow.The main motivation of the current research is the exploring the case when velocity field u is supposed to be perpendicular to the vector ∇α. Conditions for the existence of the exact solution for the aforementioned type of flows are obtained, for which non-stationary helical flow with invariant Bernoulli-function is considered.The spatial part of the pressure field of the fluid flow should be determined via Bernoulli-function, if components of the velocity of the flow are already obtained. Keywords: Navier-Stokes equations, Non-stationary helical flow, Arnold-Beltrami-Childress (ABC) flow

In this paper, absolutely new analytical ansatz for solving Kelvin–Kirchhoff equations has been presented. The aforesaid approach was formulated first in Ershkov (2017) for solving Poisson equations; furthermore, a new type of the solving procedure for Euler–Poissonequations (rigid body rotation over the fixed point) is implemented here for solving momentum equation of Kelvin–Kirchhoff. The system of equations of Kelvin–Kirchhoff problem has been explored with regard to the existence of an analytic way of presentation of the analytical solution. A new and elegant ansatz is suggested in this publication whereby, in solving, the momentum equation is reduced to a system of three linear ODEs of 1st order in regard to the three components of the velocity of the spherical particle (dependent on time t). In this premise, a proper elegant partial solution has been obtained due to the invariant dependence between temporary components of the solution. We conclude that the system of Kelvin–Kirchhoff equations has not the analytical presentation of solution (in quadratures) even in case of zero components of fluid force, influencing on the motion of the particle.

We present in this communication a new solving procedure for Kelvin&Kirchhoff equations, considering the dynamics of falling the rigid rotating torus in an ideal incompressible fluid, assuming additionally the dynamical symmetry of rotation for the rotating body, I_1 = I_2. Fundamental law of angular momentum conservation is used for the aforementioned solving procedure. The system of Euler equations for dynamics of torus rotation is explored in regard to the existence of an analytic way of presentation for the approximated solution (where we consider the case of laminar flow at slow regime of torus rotation). The second finding is associated with the fact that the Stokes boundary layer phenomenon on the boundaries of the torus is also been assumed at formulation of basic Kelvin&Kirchhoff equations (for which analytical expressions for the components of fluid torque vector {T_2, T_3} were obtained earlier). The results of calculations for the components of angular velocity should then be used for full solving the momentum equation of Kelvin&Kirchhoff system. Trajectories of motion can be divided into, preferably, 3 classes: zigzagging, helical spiral motion, and the chaotic regime of oscillations., 25 pages, 2 Figures, 1 Table; Key words: Kelvin-Kirchhoff equations, Euler equations, chaotic regime; Cite this article as: Ershkov, S., Leshchenko, D., and Giniyatullin, A., International Journal of Bifurcation and Chaos, 2021, 31(1), 2150010. https://doi.org/10.1142/S0218127421500103

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 main motivation of this research is the analytical exploration of the dynamics of asteroid rotation when it moves in elliptic orbit through Space. According to the results of Efroimsky, Frouard (2016), various perturbations (collisions, close encounters, YORP effect) destabilize the rotation of a small body (asteroid), deviating it from the initial-current spin state. This yields evolution of the spin towards rotation about maximal-inertia axis due to the process of nutation relaxation or to the proper spin state corresponding to minimal energy with a fixed angular momentum. We consider in our research the aforementioned spin state of asteroid but additionally under non-vanishing influence of the effects of non-gravitational nature (YORP effect), which is destabilizing the asteroid rotation during its motion far from giant planets. Meanwhile, new solutions for asteroid rotation dynamics in case of negligible (time-dependent) applied torques have been obtained in our development. New method for solving Euler equations for rigid body rotation is suggested; an elegant example for evolution of spin towards the rotation about maximal-inertia axis is calculated., Comment: 23 pages, 4 figures; Keywords: tidal dissipation, asteroid rotation, YORP effect; article is accepted for publication in "Acta Astronautica" (14 May 2018), DOI 10.1016/j.actaastro.2018.05.023

Before formula (1) (and, then, after formula (2)), symbol “G” should be read as “Γ”: the rigid body’s density.

Tidal interactions between Planet and its satellites are known to be the main phenomena, which are determining the orbital evolution of the satellites. The modern ansatz in the theory of tidal dissipation in Saturn was developed previously by the international team of scientists from various countries in the field of celestial mechanics. Our applying to the theory of tidal dissipation concerns the investigating of the system of ODE-equations (ordinary differential equations) that govern the orbital evolution of the satellites; such an extremely non-linear system of 2 ordinary differential equations describes the mutual internal dynamics for the eccentricity of the orbit along with involving the semi-major axis of the proper satellite into such a monstrous equations. In our derivation, we have presented the elegant analytical solutions to the system above; so, the motivation of our ansatz is to transform the previously presented system of equations to the convenient form, in which the minimum of numerical calculations are required to obtain the final solutions. Preferably, it should be the analytical solutions; we have presented the solution as a set of quasi-periodic cycles via re-inversing of the proper ultra-elliptical integral. It means a quasi-periodic character of the evolution of the eccentricity, of the semi-major axis for the satellite orbit as well as of the quasi-periodic character of the tidal dissipation in the Planet.

A new approach is developed here for resolving of the Poisson equations in case the components of angular velocity of rigid body rotation could be considered as the functions of time-parameter t only. Fundamental solution is presented by the analytical formulae in dependence on two time-dependent, the real-valued coefficients. Such coefficients as above are proved to be the solutions of mutual system of 2 Riccati ordinary differential equations (which has no analytical solution in general case). All in all, the cases of analytical resolving of Poisson equation are quite rare (according to the cases of exact resolving of the aforementioned system of Riccati ODEs). So, the system of Euler-Poisson equations is proved to have the analytical solutions (in quadratures) only in classical simplifying cases: 1) Lagrange case, or 2) Kovalevskaya case or 3) Euler case or other well-known but particular cases (where the existence of particular solutions depends on the choosing of the appropriate initial conditions).

In a novel approach for solving equations of the CR3BP first formulated in Acta Mechanica, 228(7), 2719 (2017), we present in this communication a new solving procedure for Euler-Poisson equations for solving momentum equations of the CR3BP near libration points for uniformly rotating planets having inclined orbits in the solar system with respect to the orbit of the Earth. The system of equations of the CR3BP has been explored with regard to the existence of an analytic way of presentation of the approximated solution in the vicinity of libration points. A new and elegant ansatz is suggested in this publication whereby, in solving, the momentum equation is reduced to a system of three linear ODEs of 1st order in regard to the three components of the velocity of the infinitesimal mass m (dependent on time t). In this premise, a proper elegant partial solution has been obtained due to the invariant dependence between temporary components of the solution. We conclude that the system of CR3BP equations has not the analytical presentation of solution (in quadratures) even in the vicinity of the libration points except of the generalized Jacobi integral.

In this paper, we present a review of featured works in the field of hydrodynamics with the main aim to clarify the ways of understanding the algorithms for solving the Navier-Stokes equations. Discussing the existing algorithms, approaches and analytical or semi-analytical methods, we especially note that important problems of stability for the exact solutions should be explored accordingly relate to this respect, e.g. exploring the case of non-stationary helical flows of the Navier-Stokes equations for incompressible fluids with variable (spatially dependent) coefficient of proportionality alpha between velocity and the curl field of the flow. Meanwhile, the system of Navier-Stokes equations (including continuity equation) has been successfully explored previously with respect to the existence of analytical way for presentation of non-stationary helical flows of the aforementioned type. Conditions for the stability criteria of the exact solution for such the type of flows are obtained herein in the current research, for which non-stationary helical flow with invariant Bernoulli-function is considered.

We explore the ansatz of derivation of non-stationary creeping flows for the Navier–Stokes equations in the case of incompressible flow of Newtonian fluids. In general case, such a solution should be obtained from the mixed system of 2 Riccati ordinary differential equations (in regard to time-parameter t). But we have found an elegant way to simplify it to the proper analytical presentation of creeping flow as a solution of only one Riccati ordinary differential equation, which nevertheless has no analytical solution in general case. It was a motivation to obtain the elegant approximate solution for creeping flow: - curl-free velocity field consists of the constant component U in Ox-direction, non-stationary (but negligible) component V in Oy-direction, the constant (but negligible) component W in Oz-direction. Additional component of flow velocity field (with variable curl) consists of the appropriate harmonic functions, not depending on time-parameter t. Also the suggested ansatz is illuminated by showing (on the example of creeping flow past a sphere) how it can be used to find specific solutions of current technical interest, or that it can be used to some significant advantage over some of the many existing analytical, semi-analytical, and/or numerical techniques for solution of creeping flows.

We have presented in this communication a new solving procedure for the dynamics of non-rigid asteroid rotation, considering the final spin state of rotation for a small celestial body (asteroid). The last condition means the ultimate absence of the applied external torques (including short-term effect from torques during collisions, long-term YORP effect, etc.). Fundamental law of angular momentum conservation has been used for the aforementioned solving procedure. The system of Euler equations for dynamics of non-rigid asteroid rotation has been explored with regard to the existence of an analytic way of presentation of the approximated solution. Despite of various perturbations (such as collisions, YORP effect) which destabilize the rotation of asteroid via deviating from the current spin state, the inelastic (mainly, tidal) dissipation reduces kinetic energy of asteroid. So, evolution of the spinning asteroid should be resulting by the rotation about maximal-inertia axis with the proper spin state corresponding to minimal energy with a fixed angular momentum. Basing on the aforesaid assumption (component K_1 is supposed to be fluctuating near the given appropriate constant of the fixed angular momentum), we have obtained that 2-nd component K_2 is the solution of appropriate Riccati ordinary differential equation of 1-st order, whereas component K_3 should be determined via expression for K_2., Comment: 17 pages; Keywords: tidal dissipation, asteroid rotation, angular momentum; article is accepted for publication in "Acta Astronautica" (4 May 2019); additional comments see in Appendix A2

In this paper, we present a new approach for solving Laplace tidal equations (LTE) which was formulated first in [S.V.Ershkov, A Riccati-type solution of Euler-Poisson equations of rigid body rotation over the fixed point, Acta Mechanica, 228(7), 2719 (2017)] for solving Poisson equations: a new type of the solving procedure for Euler-Poisson equations (rigid body rotation over the fixed point) is implemented here for solving momentum equation of LTE, Laplace tidal equations. Meanwhile, the system of Laplace tidal equations (including continuity equation) has been successfully explored with respect to the existence of analytical way for presentation of the solution. As the main result, the new ansatz is suggested here for solving LTE: solving momentum equation is reduced to solving system of 3 nonlinear ordinary differential equations of 1-st order in regard to 3 components of the flow velocity (depending on time t), along with the continuity equation which determines the spatial part of solution. Nevertheless, the proper elegant partial solution has been obtained due to invariant dependence between temporary components of the solution. In addition to this, it is proved here that the system of Laplace tidal equations has not the analytical presentation of solution (in quadratures) in case of nonzero fluid pressure in the Ocean, as well as nonzero total gravitational potential and the centrifugal potential (due to planetary rotation)., Comment: 19 pages, 1 figure; article is accepted for publication in "Physics of Fluids" (07 December 2018); Keywords: Laplace tidal equations, Poisson equations, Riccati equation

A new type of exact solutions for photogravitational restricted three-body problem (a case of spiral motion) is presented here. A key point is that we obtain the appropriate specific case of spiral motions from the Jacobian-type integral of motion for photogravitational restricted three-body problem (when orbit of small 3-rd body is assumed to be like a spiral). Besides, we should especially note that there is a proper restriction to the type of spiral orbital motion of small 3-rd body, which could be possible for choosing as the exact solution of equations for photogravitational restricted three-body problem. The main result, which should be outlined, is that in a case of quasi-planar orbital motion (of the small 3-rd body) the asymptotic expression for component z of motion is proved to be given by the proper elliptical integral.

A new exact solution of Euler’s equations (rigid body dynamics) is presented here. All the components of angular velocity of rigid body for such a solution differ from both the cases of symmetric rigid rotor (which has two equal moments of inertia: Lagrange’s or Kovalevskaya’s case), and from the Euler’s case when all the applied torques are zero, or from other well-known particular cases. The key features are the next: the center of mass of rigid body is assumed to be located at meridional plane along the main principal axis of inertia of rigid body, besides, the principal moments of inertia are assumed to satisfy to a simple algebraic equality. Also, there is a restriction at choosing of initial conditions. Such a solution is also proved to satisfy to Euler–Poinsot equations, including invariants of motion and additional Euler’s invariant (square of the vector of angular momentum is a constant). So, such a solution is a generalization of Euler’s case.

Previously, we have considered the equations of motion of the three-body problem in a Lagrange form (which means a consideration of relative motions of 3-bodies in regard to each other). Analyzing such a system of equations, we considered the case of small-body motion of negligible mass around the 2-nd of two giant-bodies (which are rotating around their common centre of masses on Kepler trajectories), the mass of which is assumed to be less than the mass of central body. In the current development, we have derived a key parameter that determines the character of quasi-circular motion of the small 3-rd body relative to the 2-nd body (Planet). Namely, by making several approximations in the equations of motion of the three-body problem, such the system could be reduced to the key governing Riccati-type ordinary differential equations. Under assumptions of R3BP (restricted three-body problem), we additionally note that Riccati-type ODEs above should have the invariant form if the key governing (dimensionless) parameter remains in the range 0.001-0.01. Such an amazing fact let us evaluate the forbidden zones for Moons orbits in the inner Solar system or the zones of the meanings of distances (between Moon-Planet) for which the motion of small body could be predicted to be unstable according to basic features of the solutions of Riccati-type., Comment: 10 pages, 1 Tab.; Key words: restricted three-body problem, orbits of the Moon, relative motion; Cite this article as: ERSHKOV, S.V. J Astrophys Astron (2017) 38:5. doi:10.1007/s12036-017-9425-1

Tidal interactions between Planet and its satellites are known to be the main phenomena, which are determining the orbital evolution of the satellites. We suggest in the current research to take into consideration the additional well-known effect of differential rotation which obviously takes place in the gaseous or fluid convection zone of primary giant Planet (indeed, the aforementioned effect exists even not depending on the orbital evolution of satellites around host Planet). Nevertheless, estimations for the contribution of the aforementioned effect of differential rotation in the Uranus system (including all its most massive satellites) let us exclude using such effect from calculations of mutual evolution of the eccentricity e along with the semi-major axis a for all satellites of Uranus (Planet of ice-type). It means that the Uranus can be considered in the analytic exploration of governing equations as to be the appropriate candidate for applying the modern ansatz [Efroimsky, 2015] (tidal dissipation effect depending on the tidal-flexure frequency) in regard to estimations of eccentricity e along with semi-major axis a for satellites of Uranus. We can see from the results of calculations in Section 3 that the combined system of governing equations (in the sense of combined contributions to the tidal dissipation from Uranus + from satellite) yields the really observed magnitudes of decelerations for semi-major axises of all the satellites of Uranus. Meanwhile, internal heat generation effects in the satellites (due to tidal dissipation effect) are much more than those which definitely take place in the Uranus, excepting the case of Ariel., Comment: 14 pages, 2 figures; Keywords: Tidal dissipation, differential rotation of Uranus, the Lense-Thirring effect

The generalized Euler case (rigid body rotation over the fixed point) is discussed here: - the center of masses of non-symmetric rigid body is assumed to be located at the equatorial plane on axis Oy which is perpendicular to the main principal axis Ox of inertia at the fixed point. Such a case was presented in the rotating coordinate system, in a frame of reference fixed in the rotating body for the case of rotation over the fixed point (at given initial conditions). In our derivation, we have represented the generalized Euler case in the fixed Cartesian coordinate system; so, the motivation of our ansatz is to elegantly transform the proper components of the previously presented solution from one (rotating) coordinate system to another (fixed) Cartesian coordinates. Besides, we have obtained an elegantly analytical case of general type of rotations; also, we have presented it in the fixed Cartesian coordinate system via Euler angles., Comment: 16 pages, 8 figures; Keywords: Euler equations (rigid body dynamics), Poinsot equations, Euler angles, principal moments of inertia

The existence of stationary points for the dynamical system of ABC-flow is considered. The ABC-flow, a three-parameter velocity field that provides a simple stationary solution of Euler's equations in three dimensions for incompressible, inviscid fluid flows, is the prototype for the study of turbulence (it provides a simple example of dynamical chaos). But, nevertheless, between the chaotic trajectories of the appropriate solutions of such a system we can reveal the stationary points, the deterministic basis among the chaotic behaviour of ABC-flow dynamical system. It has been proved the existence of 1 point for two partial cases of parameters {A, B, C}: 1) A = B = 1; 2) C = 1 (A^2 + B^2 = 1). Moreover, dynamical system of ABC-flow allows 3 points of such a type, depending on the meanings of parameters {A, B, C}., 10 pages, 4 figures; Keywords: Arnold-Beltrami-Childress (ABC) flow, helical flow, stationary points