Central Force Field Research Articles

Steady Motions of a Symmetric Isosceles Tetrahedron in a Central Force Field

Steady Motions of a Symmetric Isosceles Tetrahedron in a Central Force Field

Rotation of Isosceles Tetrahedron in Central Newtonian Force Field: Staude Cone

Rotation of Isosceles Tetrahedron in Central Newtonian Force Field: Staude Cone

Pedal equation and Kepler kinematics

Kepler’s laws are an appropriate topic to highlight the significance of the pedal equation in physics. There are several articles that obtain Kepler’s laws through conservation and gravitation laws. This can be shown more easily and ingeniously if one uses the pedal equation of an ellipse. In fact, the complete kinematics of a particle in an attractive central force field can be derived from one single pedal form. Though many articles use the pedal equation, the classical procedure (without proof) for obtaining the pedal equation is mentioned only in a few because the classical derivations can sometimes be lengthier and are not simple. In this paper, using elementary physics, we derive the pedal equation for all conic sections in a unique, short, and pedagogical way. Later, from the dynamics of a particle in the attractive central force field, we deduce the single pedal form, which elegantly describes all the possible trajectories. Also, for the purpose of completeness, we derive Kepler’s laws.

Analysis of the roaming trajectories from the dynamic and kinematic perspectives – A representative study of triatomic systems

We propose a method, which can be applied to the study of the roaming process in triatomic systems. Being similar to the method of dealing with the equivalent one-dimensional problem in the central force field, a triatomic roaming system is reduced to a one-dimensional problem. With this method, the mechanism of quasi-periodic and bounded motion in the roaming phenomenon can be revealed. Once the triatomic system is given, the roaming trajectories of motion can be derived, which means the evolution of roaming systems may be controlled.

Symbolic-Analytic Methods for Studying Equilibrium Orientations of a Satellite on a Circular Orbit

This paper proposes two simple methods for determining equilibrium orientations of a satellite moving in a central Newtonian force field along a circular orbit under the action of the gravitational torque. The first method uses linear algebra approaches, while the second one employs computer algebra algorithms. The equilibrium orientations of the satellite in the orbital coordinate system for given values of principal central moments of inertia are determined by the roots of a system of nonlinear algebraic equations. To determine the equilibrium solutions, the system of algebraic equations is decomposed using linear algebra methods and algorithms for Grobner basis construction. The equilibria of the satellite are determined by analyzing the real roots of the algebraic equations from the Grobner bases constructed. Using the proposed approach, it is shown that the satellite with unequal principal central moments of inertia has 24 equilibrium orientations on a circular orbit.

Constant-speed ramps for a central force field

<p style='text-indent:20px;'>We investigate the problem of determining the planar curves that describe ramps where a particle of mass <inline-formula><tex-math id="M1">\begin{document}$ m $\end{document}</tex-math></inline-formula> moves with constant-speed when is subject to the action of the friction force and a force whose magnitude <inline-formula><tex-math id="M2">\begin{document}$ F(r) $\end{document}</tex-math></inline-formula> depends only on the distance <inline-formula><tex-math id="M3">\begin{document}$ r $\end{document}</tex-math></inline-formula> from the origin. In this paper we describe all the constant-speed ramps for the case <inline-formula><tex-math id="M4">\begin{document}$ F(r) = -m/r $\end{document}</tex-math></inline-formula>. We show the circles and the logarithmic spirals play an important role. Not only they are solutions but every other solution approaches either a circle or a logarithmic spiral.

SIMULATION OF THE ROTATIVE-TRANSLATIVE MOTION OF THE DUMBBELL, WHEN THE CENTER OF ITS MASS MOVES IN A CIRCLE

A geometric model of the rotational-translational motion of a dumbbell in a central force field is proposed, provided that its center of mass moves in a circle of a given radius. The term dumbbell denotes a geometric object with two masses spaced apart at a certain distance, connected by a weightless rod (like a sports dumbbell). The idea of modeling is based on the compilation and solution of a system of Lagrange differential equations of the second kind. To compose the equations, the Lagrangian was used to describe the rotational-translational motion of a dumbbell in a central force field. The system of equations is formed with respect to five functions denoting five generalized coordinates. These coordinates connect the fixed and mobile coordinate systems that provide the rotational-translational movement of the dumbbell. The system of Lagrange differential equations of the second kind was solved numerically in the environment of the maple mathematical processor. The results obtained correspond to each of five different coordinate functions that can be plotted graphically. Also obtained are approximate descriptions of the derivatives of these functions depending on the parameters of the dumbbell and the initial conditions of its motion. These results made it possible to construct graphs of phase trajectories of functions of five coordinate coordinates, with the help of which it is possible to determine the nature of the dumbbell movement. The obtained time dependences for the functions of generalized coordinates make it possible to compose an algorithm for computer animation of the rotational-translational motion of the dumbbell. In this case, the parameters of the dumbbell and the initial conditions of its movement will be taken into account. Examples of modeling the trajectories of the centers of mass of the dumbbell weights are given. It is advisable to use the results obtained to illustrate the position of the dumbbell in space when studying its orbital dynamics. Computer animations of a dumbbell movement in zero gravity will make it possible to analyze the influence of the angular velocities of rotation on its movement. At this stage of research, it is advisable to use the results obtained as the basis for laboratory or coursework of the departments of geometric modeling and computer graphics.

Оscillations of closed conical shells with complex rotation

The paper consider a system of two closed conical shells connected by a central rigid insert rotating in opposite directions in a central force field with a constant angular velocity around the axis of symmetry of the system. The shell element is subjected to a load consisting of gravitational and inertial forces, but at large values of the angular velocity of the system, the gravitational loads can be neglected. The gyroscopic interaction between the rotational portable motion of the system and the relative elastic oscillations of the elements is a source of excitation of precession oscillations, which may be resonant or unstable. Occurring when changing the axis of orientation of the system gyroscopic moment causes the appearance of alternating stresses, which significantly affect the strength and reliability of the shells. Such problems arise in construction engineering, mechanical engineering, aircraft construction, space engineering and other sectors of the economy. The main load acting on the elements of such systems are significant centrifugal forces of inertia, which significantly affect the strength characteristics of structures. Taking into account the periodicity of the right-hand side and the coefficients of the system of resolving equations, with the help of the projection method it is possible to reduce the resolving equations to the system of ordinary differential equations, which approximately replaces the original one. The solution of the obtained system of equations makes it possible to determine the forms of oscillations and forces in a composite conical shell at various parameters of the shell and the ratios of the velocities of the shell's own rotation and the rotation of its center of mass.

Influence of the stationary and perturbed state of the central force field on the fractal characteristics of the attractor

The paper states that the known algorithms for generating and constructing fractal sets can be significantly expanded through the family of new algorithms proposed by the authors. These algorithms are based on modelling the attractors of motion of a material point in the field N of central forces in a discrete formulation. When only one of these forces is accidentally switched on at any given time, the point attractor has a strictly fractal structure. It is shown that the perturbation of one or more of the N central forces leads to a change in the structure of the attractor. Thus, the areas of the attractor Dp , controlled by the perturbed forces, with an increase in the perturbation radius, evolve to the perturbation trajectory. For biharmonic perturbations, it is shown that these subsets belong to the inner region of the 2n–point. It has been established that for small values of the perturbation radius R the parameter n → ∞, and for large values of R the parameter n → 1. For the field of central forces in the form of matrices 2*2; 3*3; 5*5 the quantitative models n(2R/B; m) are constructed and their close correlation with the perturbation parameter R, the size of the side B of the square matrix of the field of central forces and the “gravitational” parameter m is shown. It is shown that the gnoseology of the proposed algorithms originates from the wellknown algorithm of M. Barnsley, but the physical and software components are significantly improved and developed. The proposed family of algorithms allows to expand the possibilities of generating original (exclusive) fractal sets up to ~ 1040… 1050 pieces. At the same time, it is possible to control the fractal dimension, porosity, specific gravity, aerodynamic and hydraulic resistance, noise, sound and thermal insulation properties, colour of individual subregions, etc. in a wide range of values. It is shown that a significant part of such fractal sets, especially those with a high degree of symmetry, can be useful for solving problems in the field of design, ergonomics and aesthetics, for decorating buildings, clothing, footwear, haberdashery, toys, as well as for creating puzzles, IQ-tests, etc.

Mathematical Simulation of Satellite Motion with an Aerodynamic Attitude Control System Influenced by Active Damping Torques

The dynamics of a satellite moving in a central Newtonian force field in a circular orbit under the influence of aerodynamic and active damping torques depending on projections of the satellite’s angular velocity is studied. A method for determining all equilibrium positions (equilibrium orientations) of the satellite in the orbital coordinate system given the values of aerodynamic torque, damping coefficients, and the principal central moments of inertia is proposed. In the case when the axes of the coordinate system attached to the satellite coincide with the axes of the orbital coordinate system, necessary and sufficient conditions for the asymptotic stability of the corresponding zero equilibrium position are obtained using the Routh–Hurwitz criterion. The domains with satisfied asymptotic stability conditions for the zero equilibrium position are analyzed depending on various dimensionless parameters of the problem. The damping of spatial oscillations of the satellite is numerically studied for various values of aerodynamic torque and damping coefficients.

Regular Quaternion Models of Perturbed Orbital Motion of a Rigid Body in the Earth’s Gravitational Field

In this paper, we propose regular quaternion models of perturbed orbital motion of a rigid body. They do not have the features inherent in classical models when a body moves in a Newtonian gravitational field and, in the general case, when a body moves in a central force field with the potential in the form of a polynomial of negative degrees of distance to the center of attraction of the fourth-order. We also propose regularized quaternion models of the orbital motion of the body in the Earth’s gravitational field with allowance for not only the central (Newtonian), but also zonal, tesseral, and sectorial harmonics of the gravitational field potential that take the Earth’s nonsphericity into account. In these models, the negative degrees of distance to the attraction center are reduced by several orders of magnitude in terms describing the effect of the zonal, tesseral, and sectorial harmonics of the Earth’s gravitational field potential on the orbital motion of a rigid body. The main variables are the Euler parameters, the distance from the center of mass of the body to the center of attraction, the total energy of the orbital motion of the body, and the square of the absolute value for the vector of the orbital velocity moment of the body (or projection of this vector). These models use a new independent variable associated with time by a differential relation containing the square of the distance from the center of mass of the body to the center of attraction. In the case of the orbital motion of the body in the Earth’s gravitational field, in the description of which only its central and zonal harmonics are taken into account, we found the first integrals of the equations of orbital motion and proposed substitutions of variables and transformations of these equations. For studying the motion of the body, this allowed us to obtain closed systems of differential equations of lower dimension, in particular, a system of third-order equations for distance, the sine of the geocentric latitude, and the square of the absolute value of the vector of the orbital angular momentum.

Relativistic mid-wavelength infrared pulses generated in intense-laser mass-limited target interactions

Infrared spectroscopy, ultra-fast x-ray high harmonic generation, and time-resolved imaging of molecular structures benefit from the availability of intense mid-infrared wavelength pulses. Here we present a new approach to generating these, in which an intense short laser pulse is incident upon a near-critical density, spherical, mass-limited carbon target. After the laser pulse interaction, the carbon ions produced form a central force field. Plasma electrons accelerated by the laser return to the positively charged carbon target under the action of this field. The energy of these electrons is reduced and is lower than their energy in the laser field. These low-energy refluxing electrons start to oscillate with a rotating figure-of-eight motion around the positively charged carbon target and emit relativistically intense mid-infrared pulses with the wavelength in the range 1 to 4 μm.

Observable and Unobservable Mechanical Motion.

A thermodynamic approach to mechanical motion is presented, and it is shown that dissipation of energy is the key process through which mechanical motion becomes observable. By studying charged particles moving in conservative central force fields, it is shown that the process of radiation emission can be treated as a frictional process that withdraws mechanical energy from the moving particles and that dissipates the radiation energy in the environment. When the dissipation occurs inside natural (eye) or technical photon detectors, detection events are produced which form observational images of the underlying mechanical motion. As the individual events, in which radiation is emitted and detected, represent pieces of physical action that add onto the physical action associated with the mechanical motion itself, observation appears as a physical overhead that is burdened onto the mechanical motion. We show that such overheads are minimized by particles following Hamilton’s equations of motion. In this way, trajectories with minimum curvature are selected and dissipative processes connected with their observation are minimized. The minimum action principles which lie at the heart of Hamilton’s equations of motion thereby appear as principles of minimum energy dissipation and/or minimum information gain. Whereas these principles dominate the motion of single macroscopic particles, these principles become challenged in microscopic and intensely interacting multi-particle systems such as molecules moving inside macroscopic volumes of gas.

Atomistic determination on stability, cluster and microstructures in terms of crystallographic and thermo-kinetic integration of Al−Mg−Si alloys

Nanoscale-sized precipitates are directly tied to the strength and ductility of heat-treatable Al−Mg−Si alloys, thus understanding the precipitation is crucial in adjusting and optimizing their mechanical properties. In this work, precipitation behaviors of the primary strengthening phases in Al−Mg−Si alloys are investigated in light of the crystallographic, thermodynamic and kinetic analysis. The structural stability and atomic clusters of the precipitated phases are analyzed by energetic computation and central force field cluster model. The precipitation microstructures and interfacial configurations are characterized by high-resolution transmission electron microscopy. These results are then used as essential inputs for the atomistic calculations at both ground state and finite temperatures to understand the morphological formation. Accordingly, the size evolution of the primary strengthening phase is obtained within a modelling framework and compared with the microstructure data measured from experiments. Furthermore, an underlying thermo-kinetic correlation behind the precipitation is analyzed in combination with tensile properties. Our investigation offers an insight into understanding the connection among thermo-kinetics, microstructures, and mechanical properties of Al−Mg−Si alloys, and thus will provide a theoretical guidance for developing alloys with desirable strength-ductility combination.

Tension force in nanosatellite tethers systems

The use of innovative technologies in space missions has evolved considerably in the last decade. The use of large cables in space structures to connect spacecrafts and satellites with the goal to minimize cost of missions created a new field to be explored. A brief explanation will be considered about papers related to the equilibrium and stability of the movement of space systems connected by cables, known as Tether Systems. It will be presented the mathematical formulation for the system formed by two point masses connected by a tether in the central force field, in a Keplerian movement. The Lagrangian formulation was used to describe the rotational movement of the dumbbell-like system. Results of system behavior, tension and kinetic energy will be presented for two different situations, considering equal masses and different masses.

Analysis of repulsive central universal force field on solar and galactic dynamics

AbstractRecent astrophysical observations hint toward the need for an extended theory of gravity to explain puzzles presented by the standard cosmological model such as the need for dark matter and dark energy to understand the dynamics of the cosmos. This paper investigates the effect of a repulsive central universal force field on the behavior of celestial objects. Negative tidal effect on the solar and galactic orbits, like that experienced by Pioneer spacecrafts, was derived from the central force and was shown to manifest itself as dark matter and dark energy. Vertical oscillation of the sun about the galactic plane was modeled as simple harmonic motion driven by the repulsive force. The proposed universal field was used to infer the shape of dark matter halos as generated from a planar component of the universal force and to explain galactic warp, galactic halo density, and galactic rotation curves. It was found that the repulsive field addition to Newton’s gravity mimics the Yukawa potential correction employed by many current gravitational theories that modify gravity.

Stretched-deformed state of closed conical shells with complex rotation

In construction machinery, engineering, aircraft engineering, space technology and other branches of the national economy there are problems of studying the stress-strain state of plates, membranes and rotating membranes. The main load on the elements of such systems is the significant centrifugal forces of inertia, which significantly affect the strength characteristics of structures.In many cases, the axis of rotation of mechanical systems can make a turn, which leads to the emergence of not only portable and relative, but also coriolis forces of inertia, which change periodically over time. The gyroscopic interaction between the rotational motion of the system and the relative elastic oscillations of the elements is a source of excitation of precessional oscillations that can be resonant or unstable. As a result of changing the orientation axis of the system, the gyroscopic moment causes the appearance of alternating stresses, which significantly affect the strength and reliability of the structural elements.In the works [1, 3-10] the stress-strain state and oscillations of membranes, plates and shells carrying complex motion in the central force field are investigated. In this paper, the relations that determine the stress-deformed state of the system of two closed conical shells connected by a central rigid insert, which rotates with a constant angular velocity around the axis of symmetry of the system, whose center of mass moves in the central force field are given.

Analysis and experimental research on graphene's electron transparency and its application for the development of micro- and nanoelectronic devices

Theoretical and experimental data of the electron beam passage through graphene taking into account reflection of electrons are obtained. Theoretical estimates were made by considering the motion of an electron in a central force field which was modeled as a positively-charged carbon nucleus screened by the Thomas-Fermi potential. Along with the theoretical estimates, the results of experimental studies, obtained on the basis of a specially developed stand in which the electron beam was generated by irradiation of a diamond photocathode by vacuum ultraviolet, are presented. A comparison of the theoretical and experimental data showed that the experimental estimate of the passage of an electron through graphene is in good agreement with the calculated data when the first reflection is taken into account. The greatest discrepancies were observed for small values of the electron energy. The obtained results show that at the electron energies of >35 eV the electron almost freely penetrates through the graphene membrane and it corresponds to the results published earlier, and at energies < 20 eV the data differ strongly. This fact shows that in this energy range it is necessary to continue studying the graphene's transparency.Major part of the work is devoted to the use of graphene membrane's electron transparency for developing a number of unique devices and systems. These are vacuum ultraviolet radiation matrix detectors that are not sensitive to solar radiation, flat terminals with the properties of cathode ray tubes, image converter based on a vacuum emission triode, and self-excited oscillation memory cells for archival memory. It is noted that these devices and systems can be developed on the basis of existing technologies for the graphene films production.

Spin-Orbital Resonance Motion of a Satellite with Two Flexible Viscoelastic Rods in an Elliptical Orbit

The present work is aimed at studying the rotational motion of a satellite in a central force field in an elliptical orbit. A satellite is simulated by a dynamically symmetrical solid body with flexible viscoelastic rods, firmly fastened along the symmetry axis. In the absence of strain in the rods, the central ellipsoid of inertia of the satellite has a spherical shape. The averaged set of equations of perturbed motion near a 1 : 1 resonance at small eccentricities is obtained. Capture in a 1 : 1 spin-orbital resonance is substantiated.

Completely New Gravitational Physics: The Ingenious Outside-Inside Centrifuge Mechanism of Gravity Clarified

The present work implements the idea that gravity is not a fundamental force and that the observed gravitational dynamics is the result of inertial motions within a Keplerian velocity field of the Higgs Quantum Space (HQS), giving mass and ruling the inertial motions of matter-energy. The Higgs theory introduces profound changes in the current view about the nature of empty space. It introduces the idea that a real quantum fluid medium, filling up the whole of space, gives mass to the elementary particles by the Higgs mechanism, an effect analogous to the Meissner effect, giving mass to the photons within superconductors. This HQS necessarily governs the inertial motion of matter-energy and is the locally ultimate reference for rest and for motions. The HQS materializes the local Lorentz frames (LFs), turning them into local proper LFs, intrinsically stationary with respect to the local moving HQS. This HQS also necessarily is responsible for the gravitational dynamics; because it is mass that creates the gravitational fields. The observed absence of the gravitational slowing of the GPS clocks by the solar field and the absence of light anisotropy with respect to the moving earth are both obvious signatures of the true physical mechanism of gravity. These observations demonstrate that the HQS is circulating round the sun and round earth according to Keplerian velocity fields (GM/r)1/2, closely consistent respectively with the planetary motions and the orbital motion of the Moon. In this Keplerian velocity field the planets are closely stationary with respect to the local HQS and carried by the moving HQS round the sun without the need of a central force field. The Keplerian velocity field of the HQS is the only possible imaginable mechanism able to give rise to the ingenious outside-inside centrifuge mechanism of gravity that creates a central field of centrifugal forces toward the gravitational center on all matter bodies not moving strictly along direct circular equatorial orbits. The Keplerian velocity field of the HQS is shown to appropriately create all the observed effects of the gravitational fields on matter, on light and on clocks.