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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

We have explored here the case of three-dimensional non-stationary flows of helical type for the incompressible couple stress fluid with given Bernoulli-function in the whole space (the Cauchy problem). In our presentation, the case of non-stationary helical flows with constant coefficient of proportionality alpha between velocity and the curl field of flow is investigated. 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 satisfying to Laplace equation. The spatial and time-dependent parts 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. Analytical and numerical findings have been outlined including outstandung graphical presentations of various types of constructed solution in illuminating dynamical snap-shots which demonstrate developing in time the structural behaviour of topology of the aforepresented solutions., Comment: 17 pages, 5 figures; Keywords: couple stress fluid, micropolar fluid, bipolar vector Laplacian, non-stationary helical flow, Beltrami flow

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

In fluid mechanics, a lot of authors have been reporting analytical solutions of Euler and Navier-Stokes equations. But there is an essential deficiency of non-stationary solutions indeed. In our presentation, we explore the case of non-stationary flows of the Euler equations for incompressible fluids, which should conserve the Bernoulli-function to be invariant for the aforementioned system. We use previously suggested ansatz for solving of the system of Navier-Stokes equations (which is proved to have the analytical way to present its solution in case of conserving the Bernoulli-function to be invariant for such the type of the flows). Conditions for the existence of exact solution of the aforementioned type for the Euler equations are obtained. The restrictions at choosing of the form of the 3D nonstationary solution for the given constant Bernoulli-function B are considered. We should especially note that pressure field should be calculated from the given constant Bernoulli-function B, if all the components of velocity field are obtained., Comment: 18 pages, 3 figures; Keywords: Euler equations, Bernoulli-function, non-stationary solutions; article was accepted for publication in "Journal of King Saud University - Science" (20 March 2018), DOI 10.1016/j.jksus.2018.03.010

The main objective for this research was the analytical exploration of the dynamics of planar satellite rotation during the motion of an elliptical orbit around a planet. First, we revisit the results of J. Wisdom et al. (1984), in which, by the elegant change of variables (considering the true anomaly f as the independent variable), the governing equation of satellite rotation takes the form of an Abel ODE of the second kind, a sort of generalization of the Riccati ODE. We note that due to the special character of solutions of a Riccati-type ODE, there exists the possibility of sudden jumping in the magnitude of the solution at some moment of time. In the physical sense, this jumping of the Riccati-type solutions of the governing ODE could be associated with the effect of sudden acceleration/deceleration in the satellite rotation around the chosen principle axis at a definite moment of parametric time. This means that there exists not only a chaotic satellite rotation regime (as per the results of J. Wisdom et al. (1984)), but a kind of gradient catastrophe (Arnold 1992) could occur during the satellite rotation process. We especially note that if a gradient catastrophe could occur, this does not mean that it must occur: such a possibility depends on the initial conditions. In addition, we obtained asymptotical solutions that manifest a quasi-periodic character even with the strong simplifying assumptions e ~ 0, p = 1, which reduce the governing equation of J. Wisdom et al. (1984) to a kind of Beletskii equation., Comment: 15 pages; Keywords: Beletskii equation, satellite rotation, Abel ODE, gradient catastrophe. Article is published in Theoretical and Applied Mechanics Letters (7 June 2017)

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}., Comment: 10 pages, 4 figures; Keywords: Arnold-Beltrami-Childress (ABC) flow, helical flow, stationary points

In fluid mechanics, a lot of authors have been executing their researches to obtain the analytical solutions of Navier-Stokes equations, even for 3D case of compressible gas flow or 3D case of non-stationary flow of incompressible fluid. But there is an essential deficiency of non-stationary solutions indeed. We explore the ansatz of derivation of non-stationary solution for the Navier-Stokes equations in the case of incompressible flow, which was suggested earlier. In general case, such a solution should be obtained from the mixed system of 2 Riccati ordinary differential equations (in regard to the time-parameter t). But we find an elegant way to simplify it to the proper analytical presentation of exact solution (such a solution is exponentially decreasing to zero for t going to infinity). Also it has to be specified that the solutions that are constructed can be considered as a class of perturbation absorbed exponentially as t going to infinity by the null solution., Comment: 21 pages, 1 figure; this article is accepted for publication in "Rendiconti del Circolo Matematico di Palermo". Keywords: Navier-Stokes equations, non-stationary incompressible flow, Riccati ODE

Novel method for semi-analytical solving of equations of a trapped dynamics for a planetoid m4 close to the plane of mutual motion of main bodies around each other (in case of a special type of Bi-Elliptic Restricted 4-Bodies Problem) is presented. We consider here three primaries m1, m2, m3 orbiting around their center of mass on elliptic orbits which are permanently forming Lagrangian configuration of an equilateral triangle. Our aim is to obtain appoximate coordinates of quasi planar trajectory of the infinitesimal planetoid m4, when the primaries have masses equal to 1/3. Results are as follows: 1) equations for coordinates {x, y} are described by system of coupled 2-nd order ordinary differential equations with respect to true anomaly f, 2) expression for z stems from solving second order Riccati ordinary differential equation that determines the quasi-periodical oscillations of planetoid m4 not far from invariant plane {x, y, 0}., Comment: 17 pages, 7 figures; Keywords: Bi-elliptic Restricted Problem of Four Bodies, trapped dynamics of planetoid, BiER4BP, Lagrangian configuration of equilateral triangle, Riccati ODE

A novel derivation of non-stationary solutions of 3D Euler equations for incompressible inviscid flow is considered here. Such a solution is the product of 2 separated parts: - one consisting of the spatial component and the other being related to the time dependent part. Spatial part of a solution could be determined if we substitute such a solution to the equations of motion (equation of momentum) with the requirement of scale-similarity in regard to the proper component of spatial velocity. So, the time-dependent part of equations of momentum should depend on the time-parameter only. The main result, which should be outlined, is that the governing (time-dependent) ODE-system consist of 2 Riccati-type equations in regard to each other, which has no solution in general case. But we obtain conditions when each component of time-dependent part is proved to be determined by the proper elliptical integral in regard to the time-parameter t, which is a generalization of the class of inverse periodic functions., Comment: 17 pages, 8 figures; this article is accepted for publication in JKSUS

Here is presented a generalization of photogravitational restricted 3-bodies problem to the case of influence of Yarkovsky effect, which is known as reason of additional infinitesimal acceleration of a small bodies in the space (due to anisotropic re-emission of absorbed energy from the sun, other stellar sources). Asteroid is supposed to move under the influence of gravitational forces from 2 massive bodies (which are rotating around their common centre of masses on Kepler trajectories), as well under the influence of pressure of light from both the primaries. Analyzing the ODE system of motion, we explore the existense of equilibrium points for a small body (asteroid) in the case when the 2-nd primary is non-oblate spheroid. In such a case, it is proved the existence of maximally 256 different non-planar libration points in generalized photogravitational restricted 3-bodies problem when we take into consideration even a small Yarkovsky effect., Comment: 12 pages; key words: Yarkovsky effect, YORP-effect, photogravitational restricted three body problem, stability, equilibrium points, libration points, oblateness

A new type of exact solutions of the full 3 dimensional spatial Helmholtz equation for the case of non-paraxial Gaussian beams is presented here. We consider appropriate representation of the solution for Gaussian beams in a spherical coordinate system by substituting it to the full 3 dimensional spatial Helmholtz Equation. Analyzing the structure of the final equation, we obtain that governing equations for the components of our solution are represented by the proper Riccati equations of complex value, which has no analytical solution in general case. But we find one of the possible exact solution which is proved to satisfy to such an equations for Gaussian beams. Decreasing of the amplitude A of presented solution up to the zero (at the appropriate meaning of angle parameter) could be associated with the existence of an optical vortex at this point. Optical vortex (also known as a "dislocation in wave trains") is a zero of an optical field, a point of zero intensity., Comment: 15 pages, 6 figures; Keywords: Helmholtz equation, Riccati equation, Gaussian beam

A new presentation of general solution of Navier-Stokes equations is considered here. We consider equations of motion for 3-dimensional non-stationary incompressible flow. The field of flow velocity as well as the equation of momentum should be split to the sum of two components: an irrotational (curl-free) one, and a solenoidal (divergence-free) one. The obviously irrotational (curl-free) part of equation of momentum used for obtaining of the components of pressure gradient. As a term of such an equation, we used the irrotational (curl-free) vector field of flow velocity, which is given by the proper potential (besides, the continuity equation determines such a potential as a harmonic function). The other part of equation of momentum could also be split to the sum of 2 equations: - with zero curl for the field of flow velocity (viscous-free), and the proper Eq. with viscous effects but variable curl. A solenoidal Eq. with viscous effects is represented by the proper Heat equation for each component of flow velocity with variable curl. Non-viscous case is presented by the PDE-system of 3 linear differential equations (in regard to the time-parameter), depending on the components of solution of the above Heat Eq. for the components of flow velocity with variable curl. So, the existence of the general solution of Navier-Stokes equations is proved to be the question of existence of the proper solution for such a PDE-system of linear equations. Final solution is proved to be the sum of 2 components: - an irrotational (curl-free) one and a solenoidal (variable curl) components., Comment: 13 pages; Keywords: Keywords: Navier-Stokes equations, incompressible flow, Riccati equations. Appears in International Journal of Fluid Mechanics Research 06/2015

A new type of solution for the full 3+1 dimensional space-time Schroedinger equation is presented here. We consider elegant presentation of the exact solution in a spherical coordinate system, along with the assuming of separation of the two angular co-ordinates from the radial and time variables. The separation of variables follows from an assumed product form of the full potential function, which should allow us to reduce the Schroedinger Eq. to Riccati ODE in stationary case. If the angular dependence is constant, then the azimuthal dependence is linear and the polar dependence is logarithmic. With this reduction the remaining partial differential equation links the radial and time dependence which is, in effect, the standard time-dependent spherical radial Schroedinger equation. Besides, we obtain that the time-depended 2-angles solutions exist if the axis of preferential direction of wave propagation is similar to the time arrow in mechanical processes. The other possibility is the appropriate phase-shifting of the time-depended solutions. A paraxial type approximation is obtained for the solution in the limit in which the polar angle approaches zero., Comment: 11 pages, 4 figures; Keywords: Schroedinger equation, Riccati ODE, paraxial approximation

In this paper, we proceed to develop a new approach which was formulated first in Ershkov (2017) for solving Poisson equations: a new type of the solving procedure for Euler-Poisson equations (rigid body rotation over the fixed point) is suggested in the current research. Meanwhile, the Euler-Poisson system of equations has been successfully explored for the existence of analytical way for presentation of the solution. As the main result, the new ansatz is suggested for solving Euler-Poisson equations: the Euler-Poisson equations are reduced to the system of 3 nonlinear ordinary differential equations of 1-st order in regard to 3 functions; the elegant approximate solution has been obtained via re-inversion of the proper analytical integral as a set of quasi-periodic cycles. 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., Comment: 25 pages, 3 figures; Keywords: Euler equations (rigid body dynamics), Poisson equations, principal moments of inertia, Riccati equation

We introduce here in the current research the revisiting of approach to the dynamics of Sun center relative to barycenter of Solar system by using self-resulting photo-gravitational force of the Sun as the main reason of such motion. In case of slowly moving in the direction outwards with respect to the initial position of barycenter of Solar system (together with the current position of Solar system barycenter, of course) with average established velocity not less than 1050 Km/day, we should especially note that hierarchical configuration of Solar system will be preferably the same during this motion. As the main findings, we have suggested algorithm how to move towards stars using Solar self-resulting photo-gravitational force. The obvious physically reasonable assumption is that the Solar system will have been increasing its size during the evolution in a future (due to losses of the total angular momentum taking into account the tidal phenomena)., Comment: 18 pages, 3 figures, 1 Table; Keywords: barycenter of Solar system, photo-gravitational force, tidal phenomena, solar convection zone