Prikladnaâ matematika i mehanika

The article presents an analytical review of works devoted to the quaternion regularization of the singularities of differential equations of the perturbed three-body problem generated by gravitational forces, using the four-dimensional Kustaanheimo–Stiefel variables. Most of these works have been published in leading foreign publications. We consider a new method of regularization of these equations proposed by us, based on the use of two-dimensional ideal rectangular Hansen coordinates, two-dimensional Levi-Civita variables, and four-dimensional Euler (Rodrigues–Hamilton) parameters. Previously, it was believed that it was impossible to generalize the famous Levi-Civita regularization of the equations of plane motion to the equations of spatial motion. The regularization proposed by us refutes this point of view and is based on writing the differential equations of the perturbed spatial problem of two bodies in an ideal coordinate system using two-dimensional Levi-Civita variables to describe the motion in this coordinate system (in this coordinate system, the equations of spatial motion take the form of equations of plane motion) and based on the use of the quaternion differential equation of the inertial orientation of the ideal coordinate system in the Euler parameters, which are the osculating elements of the orbit, as well as on the use of Keplerian energy and real time as additional variables, and on the use of the new independent Sundmann variable. Reduced regular equations, in which Levi-Civita variables and Euler parameters are used together, have not only the well-known advantages of equations in Kustaanheimo–Stiefel variables (regularity, linearity in new time for Keplerian motions, proximity to linear equations for perturbed motions), but also have their own additional advantages: 1) two-dimensionality, and not four-dimensionality, as in the case of Kustaanheimo-Stiefel, a single-frequency harmonic oscillator describing in new time in Levi-Civita variables the unperturbed elliptic Keplerian motion of the studied (second) body, 2) slow change in the new time of the Euler parameters, which describe the change in the inertial orientation of the ideal coordinate system, for perturbed motion, which is convenient when using the methods of nonlinear mechanics. This work complements our review paper [1].

The object of research in this work is a two-mass controlled mechanical system consisting of a carrier disk rotating about its axis, fixed in space, and a carried ring connected to the disk by means of weightless elastic elements. There are no dampers in the system. The process of suppression of radial oscillations is considered from the perspective of the theory of optimal control. On sufficiently large time intervals, Newton’s numerical method is used to solve the boundary value problem of the Pontryagin’s maximum principle. The properties of phase trajectories of the system are studied depending on the initial states of the disk and ring and the number of springs in a complex model of elastic interaction. It is shown how, under certain initial conditions and parameters of the system, due to the radiality of the elastic force and the law of conservation of angular momentum, the trajectory of the center of mass of the ring tends to a circle. The specified tendency to enter the circular motion mode is not uniform and depends on the number of springs. It is shown that with a small number of elastic elements, the trajectory of the ring does not take the form of a circle, but almost complete damping of radial vibrations occurs. It has been established that with the parameters of the system considered during the numerical experiment, the control is relay with a fairly large number of switchings. In this case, the entire system is simultaneously spinning-up.

The article considers the problem of nonlinear oscillations of the motion of liquids completely filling an axisymmetric cylindrical vessel moving around a horizontal axis \(O{\kern 1pt} *{\kern 1pt} Y\). The motion of each fluid is assumed to be potential and formulated in a cylindrical coordinate system. The influence of nonlinear coefficients on the characteristics of dynamic processes during finite rotational movements of the vessel is estimated and the case of forced angular oscillations of a vessel with liquids relative to a fixed axis is considered. The main nonlinear effects associated with the rotation of the diameter of the interface of liquids are also revealed. An approximate solution of the obtained non-linear equations, found by the Bubnov-Galerkin method, was used in the article. As a result of the transformation, the amplitude-frequency characteristics and stability regions of a two-layer liquid are constructed under forced angular oscillations of a round cylindrical vessel.

In the asymptotic calculations of the first order of smallness by the dimensionless amplitude of capillary waves on the surface of charged jets of polar liquid, the effect of the relaxation effect of surface tension on the regularities of their implementation is investigated. Calculations are carried out on the model of an ideal non-compressible electrically conductive fluid. It has been shown that taking into account the effect of dynamic surface tension leads to an increase in the order of the dispersion equation, which has another damping root, which is obliged to destroy the near-surface double electric layer (destruction of the ordering of polar molecules in the near-surface layer), which undergoes electrostatic instability at sufficiently large charges (pre-breakdown in the sense of ignition of corona discharge in air). In the ideal fluid mathematical model used, the relaxation motion of the jet surface disturbances that occurs when the surface tension relaxation effect is turned on and the attenuation decrements of capillary wave motions are purely of a relaxation nature.

The ice cover is modeled by a thin elastic isotropic plate floating on the surface of a liquid of finite depth. The source of disturbances moves along the surface of the plate. The values of critical velocities at which the character of the wave disturbance changes are obtained. The angular zones in which the waves propagate are determined. The influence of the velocity of movement of the source of disturbances, the thickness of the ice plate, compression and stretching forces on the amplitudes of the waves formed is investigated.