Relative equilibria in the unrestricted problem of a sphere and symmetric rigid body

We consider the unrestricted problem of two mutually attracting rigid bodies, an uniform sphere (or a point mass) and an axially symmetric body. We present a global, geometric approach for finding all relative equilibria (stationary solutions) in this model, which was already studied by Kinoshita (1970). We extend and generalize his results, showing that the equilibria solutions may be found by solving at most two non-linear, algebraic equations, assuming that the potential function of the symmetric rigid body is known explicitly. We demonstrate that there are three classes of the relative equilibria, which we call "cylindrical", "inclined co-planar", and "conic" precessions, respectively. Moreover, we also show that in the case of conic precession, although the relative orbit is circular, the point-mass and the mass center of the body move in different parallel planes. This solution has been yet not known in the literature.Comment: The manuscript with 10 pages, 5 figures; accepted to the Monthly Notices of the Royal Astronomical Societ

Resonances and O-curves in Hamiltonian systems

We investigate the problem of the existence of trajectories asymptotic to elliptic equilibria of Hamiltonian systems in the presence of resonances.Comment: 12 page

The restricted two-body problem in constant curvature spaces

We perform the bifurcation analysis of the Kepler problem on $S^3$ and $L^3$. An analogue of the Delaunay variables is introduced. We investigate the motion of a point mass in the field of the Newtonian center moving along a geodesic on $S^2$ and $L^2$ (the restricted two-body problem). When the curvature is small, the pericenter shift is computed using the perturbation theory. We also present the results of the numerical analysis based on the analogy with the motion of rigid body.Comment: 29 pages, 7 figure

Improved ferroelectric performance of La:Hf0.5Zr0.5O2 thin films

This work was supported by Russian Science Foundation (Project. No 18-19-00527)

Non-linear stability in photogravitational non-planar restricted three body problem with oblate smaller primary

We have discussed non-linear stability in photogravitational non-planar restricted three body problem with oblate smaller primary. By photogravitational we mean that both primaries are radiating. We normalised the Hamiltonian using Lie transform as in Coppola and Rand (1989). We transformed the system into Birkhoff's normal form. Lie transforms reduce the system to an equivalent simpler system which is immediately solvable. Applying Arnold's theorem, we have found non-linear stability criteria. We conclude that $L_6$ is stable. We plotted graphs for $(\omega_1, D_2).$ They are rectangular hyperbola.Comment: Accepted for publication in Astrophysics & Space Scienc

Equilibria in the secular, non-coplanar two-planet problem

We investigate the secular dynamics of a planetary system composed of the parent star and two massive planets in mutually inclined orbits. The dynamics are investigated in wide ranges of semi-major axes ratios (0.1-0.667), and planetary masses ratios (0.25-2) as well as in the whole permitted ranges of the energy and total angular momentum. The secular model is constructed by semi-analytic averaging of the three-body system. We focus on equilibria of the secular Hamiltonian (periodic solutions of the full system), and we analyze their stability. We attempt to classify families of these solutions in terms of the angular momentum integral. We identified new equilibria, yet unknown in the literature. Our results are general and may be applied to a wide class of three-body systems, including configurations with a star and brown dwarfs and sub-stellar objects. We also describe some technical aspects of the semi-numerical averaging. The HD 12661 planetary system is investigated as an example configuration.Comment: 18 pages, 17 figures, accepted to Monthly Notices of the Royal Astronomical Societ

To the matter of the educated development of the construction industry of recreational territories

Rational creation of the consumer quality of constructions, which is taking into account some cultural, historical and other, accepted for society development paradigms enters to replace mass sustainable development of the industry of a construction in case of development of the recreational territories around megalopolises, using intellectual systems in a construction, ecological, energy-saving technologies in increase in a consumer evaluation of quality. For the residential development of the recreational territories of cities it is important to provide not only complex conditions of their development, to keep their potential for future generations, providing their social-and-economic development, to consider national and climatic features, but also to use their potential opportunities and traditions of the people inhabiting them, rationally and economically. The educated development of the construction industry is the new term entered into the use of constructors, when the consumer quality of building and structures meets national preferences taking into account traditions and advanced achievements in the use of materials, architectural and project decisions, production technologies and climatic opportunities with the minimum damage to the environment. Its accomplishment assumes enhancement and introduction of local-and-standard regulation with implementation of innovative solutions