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A new symmetry for Newtonian Dynamics is analyzed, this corresponds to going to an accelerated frame, which introduces a constant gravitational field into the system and subsequently. We consider the addition of a linear contribution to the gravitational potential \(\phi \) which can be used to cancel the gravitational field induced by going to the accelerated from, the combination of these two operations produces then a symmetry. This symmetry leads then to a Noether current which is conserved. The conserved charges are analyzed in special cases. The charges may not be conserved if the Noether current produces flux at infinity, but such flux can be eliminated by going to the CM (center of mass) system in the case of an isolated system. In the CM frame the Noether charge vanishes, Then we study connection between the Cosmological Principle and the Newtonian Dynamics which was formulated via a symmetry (Benisty and Guendelman in Mod Phys Lett A 35:2050131, 2020) of this type, but without an action formulation. Homogeneous behavior for the coordinate system relevant to cosmology leads to a zero Noether current and the requirement of the Newtonian potential to be invariant under the symmetry in this case yields the Friedmann equations, which appear as a consistency condition for the symmetry., Comment: version published in General Relativity and Gravitation, 17 pages

We examine the Poynting-Robertson (PR) effect under the influence of a logarithmic predicted as a correction to the Newtonian gravitational potential in the solar system. Our motivation emanates from the fact that today’s gravitational theories predict various corrections such that exponential, logarithmic to describe and understand the existence of dark matter, postulated to resolve discrepancies in astrophysical observations in today’s accepted theories of gravity. By saying that we simply mean that our main interest is to calculate the times that dust particles take to reach the Earth’s orbit and derive analytical expressions for the time changes of basic orbital elements in the influence of the PR effect. We use dust particle starting at a distance $r = 2.7~\text{AU}$ or approximately in the asteroid belt. In a first order perturbation treatment we obtain time solutions for the orbital elements of semi major axis, eccentricity, and mean motion, as well as expressions for the time taken for these particles to reach Earth’s orbit. We find that for circular orbits in a logarithmic potential the time taken to reach Earth is slightly larger that the elliptical orbits. Newtonian potential circular orbits require more times when compared to the logarithmic ones. Finally elliptical orbits time to reach Earth are of the same order of magnitude with a dependence on the eccentricity, density, and particle diameter.

Classical theories of thermoelasticity predict an infinite speed for thermal signals which contradicts the physical phenomena. Non-classical thermoelasticity theories have been proposed to remove this paradox. One of the non-classical theories of thermoelasticity is the Lord–Shulman theory. The main purpose of this study is to present a complete general solution for the Lord–Shulman non-classical equations of thermoelasticity for three-dimensional transversely isotropic solids. In the Lord–Shulman theory, three equations of motion accompanied with the energy equation make a set of four coupled partial differential equations. In this study, four new scalar potential functions are introduced for uncoupling the coupled displacement–temperature equations. The solution is given in terms of four scalar potential functions satisfying uncoupled partial differential equations which are of lesser complexity than the coupled displacement–temperature equations. The potential functions are governed by a wave or a repeated wave–heat equation. Completeness of the solution is proved based on retarded logarithmic and retarded Newtonian potential functions, solutions for repeated wave and heat equations, the extended Boggio’s theorem for transversely isotropic media, and the perturbation theory. It is shown that the solution presented in this paper degenerates to six special problems previously reported in the literature.

In this note, we examine the scattering of two identical fermions in theories where fermionic fields minimally coupled to higher derivative gravity. In particular, we consider the extension of general relativity with $R^2$ corrections or non-local terms. We expand the action of fermions around the flat space background and obtain two fermion-one graviton vertex. Then, by considering the scattering amplitude of two fermions, we calculate the non-relativistic limit and that obtain the potential for two fermion-fermion interaction which would be the usual Newtonian potential corrected with a Yukawa-like term. At the end, we briefly discuss the astronomical effects of such Yukawa-like potential by computing the gravitational pressure of a spherical star and use it for a white dwarf to obtain quantum corrections of Chandrasekhar radius., 5 pages with 2 figures

Recently, a nonlocal generalized uncertainty principle was derived based on the new notion of quantum acceleratum operator within the framework of nonlocal-maximal quantum mechanics. In this study, we discuss some of its properties and some of its implications in Newtonian gravity theory and Verlinde’s entropic holographic approach. A number of features were revealed; in particular, the emergence of a logarithmic correction to the gravitational Newtonian potential, a minimum energy and a minimum mass which depend on the gravitational coupling constant. Based on the concept of holographic principle, Verlinde’s conjecture and equipartition rule, a quantized Newton’s force law of gravity for a particle of mass m gravitating around a Planck mass is derived.

Ultra-cold atom experiment in space with microgravity allows for realization of dilute atomic-gas Bose-Einstein condensate (BEC) with macroscopically large occupation number and significantly long condensate lifetime, which allows for a precise measurement on the shape oscillation frequency by calibrating itself over numerous oscillation periods. In this paper, we propose to measure the Newtonian gravitational constant via ultra-cold atom BEC with shape oscillation, although it is experimentally challenging. We also make a preliminary perspective on constraining the modified Newtonian potential such as the power-law potential, Yukawa interaction, and fat graviton. A resolution of frequency measurement of $(1-100)\,\mathrm{nHz}$ at most for the occupation number $10^9$, just one order above experimentally achievable number $N\sim10^6-10^8$, is feasible to constrain the modified Newtonian potential with Yukawa interaction greatly beyond the current exclusion limits., Comment: v1, 10 pages, 3 figures; v2, references added, discussion extended for enlarged parameter space of Yukawa interaction, version accepted by EPJC

We study nondegeneracy of ground states of the Hartree equation $$ -\Delta u+u=(I_{2}\ast u^2)u\quad\mbox{ in }\mathbb R^n $$ where $n=3,4,5$ and $I_2$ is the Newton potential. As an application of the nondegeneracy result, we use a Lyapunov-Schmidt reduction argument to construct multiple semiclassical solutions to the following Hartree equation with an external potential $$-\varepsilon^2\Delta u+u+V(x)u=\varepsilon^{-2}(I_{2}\ast u^2)u\quad \mbox{ in }\mathbb R^n.$$, Comment: The multipole expansion section is revised. Some refrefeces updated

We study the motion of a secondary celestial body under the influence of a corrected gravitational potential in a modified Newtonian dynamics scenario. Furthermore we look within the Milky-way where the first correction to the potential results from a modified Poisson equation, and includes two mew terms one of which is of the form $\ln(r/r_{\max})$ and the other is associated with the cosmological constant lambda $\varLambda $ added to the Newtonian potential. The regions of influence of the two potentials are associated with regions of interested bounded by the conditions $r < r_{\max}$ for the Newtonian potential, $r > r_{\max}$ the logarithmic correction to the potential relating to the term $(\nabla \phi )^{2}$ in the Poisson equation for the gravitational field that has matter density $\rho $ , and finally, the domain where $r \gg r_{\max}$ the potential scales as $c^{2}\varLambda r^{2}$ and the cosmological constant lambda dominates. Next using an average disturbing potential we integrate Lagrange’s planetary equations and we obtain analytical expressions for the average time rates of change of the orbital elements using our sun as an example. We find that both dark matter and cosmological constant affect only the argument of the perigalaktikon point as well as the mean anomaly.

We explore the dynamics and stability of the two-body problem by modifying the Newtonian potential with the Yukawa potential. This model may be considered a theory of modified gravity; where the interaction is not simply the Kepler solution for large distance. We investigate stability by deriving the Jacobian of the linearized matrix equation and evaluating the eigenvalues of the various equilibrium points calculated during the analysis. The subcases of a purely Yukawa and purely Newtonian potential are also explored.

In this paper, the Newtonian potential is modified by using the continued fraction potential. A simple formula for the modified potential is constructed, which is easy and powerful in practical uses. We investigate that the new parameter of the modified potential may play the same role of the zonal harmonic coefficients in the potential of an oblate body. The periodic solution of the two-body problem within frame of the central body has the modified potential is found by KB averaging method. The difference of the perigee vectors is also estimated at subsequent orbits. We underline that the model studied in this work could be used to obtain a harmony between the theoretical predictions and the observational evidences.

We propose and study elements of potential theory for the sub-Laplacian on homogeneous Carnot groups. In particular, we show the continuity of the single-layer potential and establish Plemelj-type jump relations for the double-layer potential. As a consequence, we derive a formula for the trace on smooth surfaces of the Newton potential for the sub-Laplacian. Using this, we construct a sub-Laplacian version of Kac’s boundary value problem.

We prove the existence and study properties of the Green function of the unit ball for the Dunkl Laplacian $\Delta_k$ in $\mathbb{R}^d$. As applications we derive the Poisson-Jensen formula for $\Delta_k$-subharmonic functions and Hardy-Stein identities for the Poisson integrals of $\Delta_k$. We also obtain sharp estimates of the Newton potential kernel, Green function and Poisson kernel in the rank one case in $\mathbb{R}^d$. These estimates contrast sharply with the well-known results in the potential theory of the classical Laplacian., Comment: 25 pages

We give universal inequalities for the norm of certain potential operators in terms of the least non-zero Neumann eigenvalue.

We rigorously justify the mean-field limit of anN-particle system subject to Brownian motions and interacting through the Newtonian potential in$${\mathbb {R}}^3$$R3. Our result leads to a derivation of the Vlasov–Poisson–Fokker–Planck (VPFP) equations from the regularized microscopicN-particle system. More precisely, we show that the maximal distance between the exact microscopic trajectories and the mean-field trajectories is bounded by$$N^{-\frac{1}{3}+\varepsilon }$$N-13+ε($$\frac{1}{63}\le \varepsilon 163≤ε<136) with a blob size of$$N^{-\delta }$$N-δ($$\frac{1}{3}\le \delta 13≤δ<1954-2ε3) up to a probability of$$1-N^{-\alpha }$$1-N-αfor any$$\alpha >0$$α>0. Moreover, we prove the convergence rate between the empirical measure associated to the regularized particle system and the solution of the VPFP equations. The technical novelty of this paper is that our estimates rely on the randomness coming from the initial data and from the Brownian motions.

Pseudo-Newtonian Potential has always been a useful tool to discuss the motion of a particle in space-time to avoid the tedious and nearly impossible nonlinear computations coming from the field equations of general relativity. Mukhopadhyay, in 2002, has introduced such a pseudo-Newtonian potential for rotating Kerr black hole which is efficient enough to replicate the scenario of the classical mechanics. But there was no such model to explain the dark energy realm. In 2016 S Ghosh introduced a Lagrangian for such rotating black hole embedded in quintessence. in this article we obtained a pseudo-Newtonian force for this new black hole solution embedded quintessence. This paper introduces a simple computational scheme to evaluate a pseudo-Newtonian force for any space-time metric. This model possesses at most $4.95 \%$ error corresponding to general relativistic results. Since we took a popular agent of dark energy, i.e., quintessence into account, this is a general form of pseudo-Newtonian force to explain late time accelerating universe. In this paper, it also has been discussed about the difference between the pseudo-Newtonian force with and without dark energy effect. This paper also explains the natures of our present universe and its fate(locally around a black hole when repulsive negative pressure of dark energy is taken into account)., Comment: 108 figures

We show that the model of superfluid dark matter developed in Refs.~\cite{Khoury:2014tka,Berezhiani:2015bqa,Berezhiani:2015pia}, which modifies the Newtonian potential and explains the galactic rotational curves, can be unitarized by the formation of classical configurations in the scattering amplitudes. The classicalization mechanism may also trigger the formation of the superfluid state from the early to the late Universe., Comment: We add more detailed calculations on the classicalon production in the early Universe. Main results and conclusions are the same

We consider the minimal energy problem on the unit sphere \({\mathbb {S}}^2\) in the Euclidean space \({\mathbb {R}}^3\) immersed in an external field Q, where the charges are assumed to interact via Newtonian potential 1/r, r being the Euclidean distance. The problem is solved by finding the support of the extremal measure, and obtaining an explicit expression for the equilibrium density. We then apply our results to an external field generated by a point charge, and to a quadratic external field.

We characterize functions $$V\le 0$$ for which the heat kernel of the Schrodinger operator $$\Delta +V$$ is comparable with the Gauss–Weierstrass kernel uniformly in space and time. In dimension 4 and higher the condition turns out to be more restrictive than the condition of the boundedness of the Newtonian potential of V. This resolves the question of V. Liskevich and Y. Semenov posed in 1998. We also give specialized sufficient conditions for the comparability, showing that local $$L^p$$ integrability of V for $$p>1$$ is not necessary for the comparability.

The common nature of the dark sector - dark energy and dark matter - as shown in [1] follows readily from the consideration of generalized Newtonian potential as a weak-field General Relativity. That generalized potential satisfying the Newton's theorem on the equivalence of sphere's gravity and that of a point-mass located in its center, contains an additional constant which along with the gravitational constant is able to explain quantitatively both the dark energy (cosmological constant) and dark matter. So, gravity is defined not by one but two fundamental constants. We show that, the second constant is dimensional-independent and matter-uncoupled and hence is even more universal than the gravitational constant, thus affecting the strategy of observational studies of dark energy and of the search of dark matter., Comment: To appear in Eur Phys J C; 5 pages

We prove that, for generic (open and dense) values of the masses, the Newtonian potential function of the collinear N-body problem has critical values when restricted to a fixed inertia level. In particular, we prove that for generic values of the masses, there is only one global minimal Moulton configuration.

The existence and stability of circular orbits (CO) in static and spherically symmetric (SSS) spacetime are important because of their practical and potential usefulness. In this paper, using the fixed point method, we first prove a necessary and sufficient condition on the metric function for the existence of timelike COs in SSS spacetimes. After analyzing the asymptotic behavior of the metric, we then show that asymptotic flat SSS spacetime that corresponds to a negative Newtonian potential at large $r$ will always allow the existence of CO. The stability of the CO in a general SSS spacetime is then studied using the Lyapunov exponent method. Two sufficient conditions on the (in)stability of the COs are obtained. For null geodesics, a sufficient condition on the metric function for the (in)stability of null CO is also obtained. We then illustrate one powerful application of these results by showing that an SU(2) Yang-Mills-Einstein SSS spacetime whose metric function is not known, will allow the existence of timelike COs. We also used our results to assert the existence and (in)stabilities of a number of known SSS metrics., 17 pages, no figure; references adjusted; error corrected

We consider a Yukawa-type gravitational potential combined with the Poynting-Robertson effect. Dust particles originating within the asteroid belt and moving on circular and elliptic trajectories are studied and expressions for the time rate of change of their orbital radii and semimajor axes, respectively, are obtained. These expressions are written in terms of basic particle parameters, namely their density and diameter. Then, they are applied to produce expressions for the time required by the dust particles to reach the orbit of Earth. For the Yukawa gravitational potential, dust particles of diameter $10^{ - 3}$ m in circular orbits require times of the order of $8.557 \times 10^{6}$ yr and for elliptic orbits of eccentricities $e =0.1, 0.5$ require times of $9.396 \times 10^{6}$ and $2.129 \times 10^{6}$ yr respectively to reach Earth’s orbit. Finally, various cases of the Yukawa potential are studied and the corresponding particle times to reach Earth’s are derived per case along with numerical results for circular and various elliptical orbits.

Studies of near periodic patterns in many self-organizing physical and biological systems give rise to a nonlocal geometric problem in the entire space involving the mean curvature and the Newtonian potential. One looks for a set in space of the prescribed volume such that on the boundary of the set the sum of the mean curvature of the boundary and the Newtonian potential of the set, multiplied by a parameter, is constant. Despite its simple form, the problem has a rich set of solutions and its corresponding energy functional has a complex landscape. When the parameter is sufficiently large, there exists a solution that consists of two tori: a larger torus and a smaller torus. Due to the axisymmetry, the problem is formulated on a half plane. A variant of the Lyapunov-Schmidt procedure is developed to reduce the problem to minimizing the energy of the set of two exact tori, as an approximate solution, with respect to their radii. A re-parameterization argument shows that the double tori so obtained indeed solves the equation of mean curvature and Newtonian potential. One also obtains the asymptotic formulae for the radii of the tori in terms of the parameter. This double tori set is the first known disconnected solution.

This paper continue the study of the generalized Lennard-Jones potential started in Barbosu et al. (J Math Chem 49(9):1961–1975, 2011) for a more general situation. More precisely we study the two-body problem with generalized Lennard-Jones potential in an anisotropic space. We will show that the set of initial conditions leading to collisions and ejections have positive measure. We also study the capture and escape solutions in the zero-energy case using the infinity manifold. We also show that the flow on the zero energy manifold of a two-body problem given by the sum of the Newtonian potential and the two anisotropic perturbations corresponding to the generalized Lennard-Jones potential is chaotic.

The problem of determination of the stationary temperature field in a body with plane sheet of heat sources is reduced to the solution of an integral equation of the first kind. A method for the determination of the set of solutions to this equation is proposed. The components of the vector of elastic displacements and the components of the tensor of temperature stresses are found according to the known temperature field by using the equations of thermoelasticity. We study the distributions of thermal displacements and stresses for any axially symmetric distribution of temperature in a circular domain. Numerous elements of contemporary engineering constructions operate under conditions of nonuniform heating that promote the formation of temperature gradients and thermal stresses. For the comprehensive analysis of the strength of structures, it is necessary to know the levels of thermal stresses and the character of their action. The investigation of the thermoelastic state of heat-generating elements is of high importance for practical purposes [9]. The presence of thin heat-generating inclusions in a body leads to the formation of tensile or compressive stresses. In the first case, the action of these stresses, together with the action of the other factors, may lead to crack initiation and propagation. In the second case, the action of stresses may decrease the intensity of stresses in the vicinity of the existing crack under force loading and promote the formation of conditions inhibiting the process of crack growth [10]. In the investigation of the stressed state of a body with thermally active cracks (with temperature or heat fluxes given on these cracks), the determination of stresses acting at the sites of the cracks is an intermediate stage and then the stresses acting upon the crack surfaces are removed. The major part of investigations of the stress-strain state of bodies with circular thermally active or heatinsulated cracks are performed for axially symmetric problems by the method of integral Hankel transformation and dual integral equations [11–16]. In [3], a method of two-dimensional boundary integral equations based on the theory of the Newton potential is developed for the solution of three-dimensional problems of stationary heat conduction and thermoelasticity for bodies with cracks. According to this method, the densities of potentials in the problem of heat conduction are the intensities of heat sources at the sites of the cracks determined, for given temperatures on the cracks, from the integral equations with polar kernels. The temperature field and stressed state of a body with given temperature or heat flow in a circular domain described by polynomials of the third degree are investigated in [4]. In the axially symmetric case, for polynomials of any degree, this problem is studied in [5]. According to the classic statement of the problems of stationary heat conduction with thermally active inclusions [4, 5], the presence of heat sources is postulated solely in the domain of the inclusion, which leads to singular distributions of the heat flows on its edges.

A configuration whose density profile coincides with the Newtonian potential for spiral galaxies is constructed from a 4D isotropic metric plus extra dimensional components. A Miyamoto-Nagai ansatz is used to solve Einstein equations. The stable rotation curves of such system are computed and, without fitting techniques, we recover with accuracy the observational data for flat or not asymptotically flat galaxy rotation curves. The density profiles are reconstructed and compared to that obtained from the Newtonian potential., Comment: 10 pages, 10 figures, submitted to Brazilian Journal of Physics

We use a non-singular potential that appears in the literature under the influence of which the Poynting- Robertson effect is studied. For that, dust particles originat- ing within the asteroid belt are used, in circular and elliptic orbits, and expressions for the semimajor axis as a function of time are obtained. The derived expressions are written in terms of the two basic dust particle parameters, namely the density and the diameter. In both cases, we obtain expres- sions for the time that the dust particles take to reach the orbit of Earth under the action of the non-singular poten- tial and solar radiation. For the non-singular potential, dust particles of diameter 10 −3 m in circular and elliptical orbits require times of the order of 4.058 × 10 7 and 2.823 × 10 7 y to reach the orbit of the Earth respectively. Finally, the de- rived expressions and numerical results are compared with those of the Newtonian potential.

The paper deals with the Sturm-Liouville operator with singular potential. We assume that the potential is a sum of an a priori known distribution from a certain class and an unknown sufficiently smooth function. The inverse problem is to recover the operator using zeros of eigenfunctions (nodes) as an input data. For this inverse problem we obtain a procedure for constructing the solution.

The paper is concerned with the evaluation of the convolution integral $${\int_{\mathbb{R}^d}\frac{1}{\left\Vert x-y\right\Vert} f(y){\rm d}y}$$ in d dimensions (usually d = 3), when f is given as piecewise polynomial of possibly large degree, i.e., f may be considered as an hp-finite element function. The underlying grid is locally refined using various levels of dyadically organised grids. The result of the convolution is approximated in the same kind of mesh. If f is given in tensor product form, the d-dimensional convolution can be reduced to one-dimensional convolutions. Although the details are given for the kernel $${{1 / \left \Vert x \right\Vert,}}$$ the basis techniques can be generalised to homogeneous kernels, e.g., the fundamental solution $${{const\cdot\left\Vert x\right\Vert ^{2-d}}}$$ of the d-dimensional Poisson equation.

A generalized uncertainty principle admitting a minimal measurable length contains a parameter of which the numerical value needs to be fixed. In fact, the application of the Generalized Uncertainty Principle (GUP) to some quantum mechanical problems offers different values for the upper bound of the GUP dimensionless parameter \(\alpha_{0}\). In this work, by applying a GUP which is linear and quadratic in the \(\Delta P\) correction to Newton’s law of gravity, and then using the stability condition of the circular orbits of the planets, we propose an upper bound for \(\alpha_{0}\). By using the astronomical data of the Solar System objects, a new and severe constraint on the upper bound of the parameter \(\alpha_{0}\) is derived. Also, using the modified Newtonian potential, inspired by a GUP which is linear and quadratic in \(\Delta P\), we investigate the possibility of measuring the relevant parameter \(\alpha_{0}\) through observables provided by the Galileo Navigation Satellite System.

We present the analytical solution to the linear evolution equation of a one component Friedmann perturbation using an equation of state of the form p = (1/3)μσ2(t), where μ is the mass density and σ(t) is the root mean square (rms) velocity in the matter dominated epoch. It is assumed that this rms velocity depends only on the time coordinate and decreases as 1/a, a being the expansion factor of the Friedmann background. The evolution equations are written for scales below the horizon using the longitudinal gauge. The general solution, in the coordinate space, of the evolution equation for the scalar mode is obtained. In the case of spherical symmetry, this solution is expressed in terms of unidimensional integrals of the initial conditions: the initial values of the Newtonian potential and its first time derivative. This perfect fluid solution is a good approximation to the evolution of warm dark matter perturbations obtained by solving the Vlasov’s equation for collisionless particles.

In this paper, we consider the problem of motion of a point on a two-dimensional sphere and pseudosphere (with metric of constant curvature) in the field generated by two attracting centers. This problem is a natural analog of the classical plane two-center problem integrated even by Euler. However, the integrability of similar problems on the sphere and the Lobachevskii plane was established only recently. Therefore, the topological properties of these problems are poorly studied as yet. The present paper fills this gap. We also study the Lagrange problem on the pseudosphere under the action of the Newtonian attraction of the stationary center and one more force constant in magnitude and direction. This problem is a limit case of the two-center problem. In the process of passing to the limit, the second center moves at infinity in the direction of the propulsion force (moreover, its mass grows so that the constancy of the propulsion is ensured, i.e., proportionally to the squared distance). This problem was considered by Lagrange for the first time and reduces to quadratures. The question on gravitational interactions in spaces of constant curvature was posed for the first time by N. N. Lobachevskii, who studied generalizations of the attraction law to the space of constant negative curvature. This theme was further developed in the work of N. E. Zhukovskii [18] on the motion of a pseudospheric (“plane”) plate on the Lobachevskii plane. On a three-dimensional sphere (of constant curvature 1), one also considers an analog of the Newtonian potential generated by a gravitational center, which in this case is cotan θ, where the angle θ is equal to the distance to the center. The integrability of the Kepler problem on the three-dimensional sphere S3 was proved by E. Schrodinger [12]. Schrodinger considered the motion of an electron in the potential field of the nucleus of a hydrogen-like atom from the standpoint of quantum mechanics. Three classical Kepler laws have natural generalizations for spaces of constant curvature. For the Lobachevskii space and for the sphere, these generalizations were formulated by A. N. Chernikov [5] and V. V. Kozlov [8]. In particular, any finite orbit here is an ellipse in one of whose foci the gravitational center (the Sun) is located. Along with this, V. V. Kozlov and O. A. Kharin proved the integrability of the problem on the motion of a material particle on the sphere S2 in the field of two stationary Newtonian centers [9]. In [14–17], the problem of two centers on the three-dimensional sphere S3 and in the three-dimensional Lobachevskii space H3 and also the Lagrange problem (the limit case of the two-center problem when one of the centers is moved off at infinity) in the Lobachevskii space H3 were studied. In these works, the integrabiity of the problems is demonstrated, bifurcation diagrams are constructed, and the classification of regions of feasible motion is carried out. The first step in studying these problems consists of reducing the problem from the three-dimensional to the two-dimensional case, i.e., reducing the number of degrees of freedom. The next step consists of passing to convenient coordinates (in the problems under consideration, for instance, to spherical-conical and pseudo-spherical-conical coordinates which are defined as roots of certain rational functions) in order to reduce the system to the Liouville form.

The tetrad and the torsion fields due to a rapidly rotating massive object are found. The motion of a spin particle in the Weitzenbock spacetime is studied. It is shown that the axial-vector torsion is the entity responsible for the gravitomagnetic component of the gravitational field.The influences of the quadrupole moment of the rapidly rotating object on the motion of the particle are discussed. It is pointed out that the influences of the quadrupole moment are negligible for Kerr black holes, but are as important as that of the Newtonian potential for a rapidly rotating neutron star.

The determination of the local gravity field by means of the point mass inversion method can be performed as an alternative to conventional numerical methods, such as the least-squares collocation. Based on the first derivative of the inverse-distance Newtonian potential for the representation of the gravity anomaly data, it is possible to compute any wavelength component of the geoid in planar approximation with sufficient accuracy. In order to exemplify the theoretical concept, two applications are presented of the computation of two different wavelength components of the geoid, the long wavelength component in a local solution and the short wavelength component in a regional solution. The results are compared with corresponding least-squares collocation solutions, using a global geopotential model to remove and to restore the long wavelength component.

In boundary element methods, the evaluation of the weakly singular integrals can be performed either a) numerically, b) symbolically, i.e., by explicit expressions, or c) in a combined manner. The explicit integration is of particular interest, when the integrals contain the singularity or if the singularity is rather close to the integration domain. In this paper we describe the explicit expressions for the sixfold volume integrals arising for the Newton potential, i.e., for a 1/r integrand. The volume elements are axi-parallel bricks. The sixfold integrals are typical for the Galerkin method. However, the threefold integral arising from collocation methods can be derived in the same way.

We review the derivation of the metric for a spinning body of any shape andcomposition using linearized general relativity theory (LGRT), and also obtainthe same metric using a transformation argument. The latter derivation makes itclear that the linearized metric contains only the Eddington α and γ parameters,so no new parameter is involved in frame-dragging or Lense—Thirring effects.We then calculate the precession of an orbiting gyroscope in a general weakgravitational field described by a Newtonian potential (the gravitoelectric field)and a vector potential (the gravitomagnetic field). Next we make a multipoleanalysis of the potentials and the precession equations, giving all of these interms of the spherical harmonics moments of the density distribution. The analysisis not limited to an axially symmetric source, although the Earth, which is themain application, is very nearly axisymmetric. Finally, we analyze the precessionin regard to the Gravity Probe B (GP-B) experiment, and find that the effect ofthe Earth's quadrupole moment (J2) on the geodetic precession is large enoughto be measured by GP-B (a previously known result), but the effect on theLense—Thirring precession is somewhat beyond the expected GP-B accuracy.

Let u be the Newtonian potential of a real analytic distribution in an open set Ω. In this paper we assume u is analytically continuable from the complement of Ω into some neighborhood of a point x0 ∈ ∂Ω, and we study conditions under which the analytic continuation implies that ∂Ω is a real analytic hypersurface in some neighborhood of x0.

The wave type field equation $\square \vt^a=\la \vt^a$, where $\vt^a$ is a coframe field on a space-time, was recently proposed to describe the gravity field. This equation has a unique static, spherical-symmetric, asymptotically-flat solution, which leads to the viable Yilmaz-Rosen metric. We show that the wave type field equation is satisfied by the pseudo-conformal frame if the conformal factor is determined by a scalar 3D-harmonic function. This function can be related to the Newtonian potential of classical gravity. So we obtain a direct relation between the non-relativistic gravity and the relativistic model: every classical exact solution leads to a solution of the field equation. With this result we obtain a wide class of exact, static metrics. We show that the theory of Yilmaz relates to the pseudo-conformal sector of our construction. We derive also a unique cosmological (time dependent) solution of the described type., Comment: Latex, 17 pages

We present a fast/parallel Poisson solver on disks, based on efficient evaluation of the exact solution given by the Newtonian potential and the Poisson integral. Derived from an integral formulation, it is more accurate and simpler in parallel implementation and in upgrading to a higher order algorithm, than an algorithm which solves the linear system obtained from a differential formulation.

The main result is an asymptotic formula for a solution to the conjugation problem for the Navier-Stokes equations describing the slow motion of two immiscible liquids such that one of them occupies a bounded domain Ω1 ⊂ ℝ3, whereas the other occupies the exterior domain Ω2=ℝ4∖Ω. Such a formula was obtained for a solution to the exterior problem with sticking conditions on the boundary in the works of Fischer, Hsiao, and Wendland. The result obtained is applied to the proof of the solvability of a free-boundary problem describing a uniform drop in an infinite liquid. Bibliography: 10 titles.

The problem of determining the effect of the media interface in a piecewise-homogeneous body on the stress concentration in the vicinity of cracks located in one of the half-spaces is reduced to a system of two-dimensional singular equations of Newtonian potential type. We study the effect of the relations of the elastic constants of the materials of a body composed of two half-spaces on the stress intensity factors in the vicinity of a crack. We study different variants of the crack location relative to the interface.

The main problem with the standard gauge theory of the Poincare group realized as a subgroup of GL(5, R) is that fields, whose physical sense is unclear, appear in connection with the non-Lorentz symmetries. Here, the Poincare fields are treated as new Yang-Mills-type tensor fields and gravity is treated as a Higgs-Goldstone field. In this case, the effective metric tensor for matter is a hybrid of two tensor fields. In the linear approximation, the massive translation gauge field gives the Yukawa-type correction to the Newtonian potential. Also, corrections to the standard Einstein post-Newtonian formulas for light deflection and radar echo delay are obtained.

The problem of two bodies with a central-force potential is treated by means of a relativistic Hamiltonian multi-time formalism. It is shown that only a Newtonian potential is compatible within this formalism, thus stating an exception of the so-called no-interaction theorems. The relativistic meaning of the Runge-Lenz vector of the classical Kepler problem is revealed.

We propose a new alternative gauge for the Einstein equations instead of the de Donder gauge, which allows in the limit of weak fields a straightforward integration of these equations. The Newtonian potential plays a new and interesting role in this framework. The calculations are demonstrated explicitly for two simple astrophysical models.

In the framework of the relativistic theory of gravity, the equation of motion for a spherically-symmetric singular shell is derived and integrated in the first approximation of the Newton potential U = m/r. We use the covariant energy-momentum conservation law for matter in the effective Riemannian space and, independently, the energy-momentum conservation law for the matter + gravity system in Minkowski space. For the problem under consideration, we show the equivalence of our approach to the classical formalism of singular shells.

We consider new spectral relations for harmonic polynomials of two variables. The basic relations of this type are the integral relations in a disk, which make it possible to use these polynomials to construct the solution of certain singular equations of Newtonian potential type, of both first and second kind, given on the disk. By use of these results we determine the eigenvalues of certain two-dimensional integral equations of second kind.

We propose a method of approximate solution of problems of elasticity theory for a half-space with protuberances based on the use of jump conditions in the stresses and displacements at a thin elastic element. The problem of determining the stresses reduces to a system of two-dimensional integral equations of Newtonian potential type for determining the contact stresses between the protuberances and the half-space. We consider the case when the elastic characteristics of the material of the protuberances are different from the material of the half-space.

In this paper, we consider a de Sitter thick brane model in a pure geometric Weyl integrable five-dimensional space-time, which is a generalization of Riemann geometry and is invariant under a so-called Weyl rescaling. We find a solution of this model via performing a conformal transformation to map the Weylian structure into a familiar Riemannian one with a conformal metric. The metric perturbations of the model are discussed. For gravitational perturbation, we get the effective modified P$\ddot{\text{o}}$schl-Teller potential in corresponding Schr$\ddot{\text{o}}$dinger equation for Kaluza-Klein (KK) modes of the graviton. There is only one bound state, which is a normalizable massless zero mode and represents a stable 4-dimensional graviton. Furthermore, there exists a mass gap between the massless mode and continuous KK modes. We also find that the model is stable under the scalar perturbation in the metric. The correction to the Newtonian potential on the brane is proportional to $e^{-3 r \beta/2}/r^2$, where $\beta$ is the de Sitter parameter of the brane. This is very different from the correction caused by a volcano-like effective potential., Comment: 24 pages, 13 figures, published version