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The regular polygon problem of ( $N+1$ ) bodies deals with the dynamics of a small body, natural or artificial, in the force field of $N$ big bodies, the $\nu=N-1$ of which have equal masses and form an imaginary regular $\nu $ -gon, while the $N$ th body with a different mass is located at the center of mass of the system. In this work, instead of considering Newtonian potentials and forces, we assume that the big bodies create quasi-homogeneous potentials, in the sense that we insert to the inverse square Newtonian law of gravitation an inverse cube corrective term, aiming to approximate various phenomena due to their shape or to the radiation emitting from the primaries. Based on this new consideration, we apply a general methodology in order to investigate by means of the zero-velocity surfaces, the regions where 3 $D$ motions of the small body are allowed, their evolutions and parametric variations, their topological bifurcations, as well as the existing trapping domains of the particle. Here we note that this process is definitely a fundamental step of great importance in the study of many dynamical systems characterized by a Jacobian-type integral of motion in the long way of searching for solutions of any kind. [ABSTRACT FROM AUTHOR]

In the present paper, using the first-order approximation of the $n$ -body Lagrangian (derived on the basis of the post-Newtonian gravitational theory of Einstein, Infeld, and Hoffman), we explicitly write down the equations of motion for the planar circular restricted three-body problem in the Solar system. Additionally, with some simplified assumptions, we obtain two formulas for estimating the values of the mass-distance and velocity-speed of light ratios appropriate for a given post-Newtonian approximation. We show that the formulas derived in the present study, lead to good numerical accuracy in the conservation of the Jacobi constant and almost allow for an equivalence between the Lagrangian and Hamiltonian approaches at the same post-Newtonian order. Accordingly, the dynamics of the system is analyzed in terms of the Poincaré sections method and Lyapunov exponents, finding that for specific values of the Jacobi constant the dynamics can be either chaotic or regular. Our results suggest that the chaoticity of the post-Newtonian system is slightly increased in comparison with its Newtonian counterpart. [ABSTRACT FROM AUTHOR]

The Copenhagen problem is a well-known case of the famous restricted three-body problem. In this work instead of considering Newtonian potentials and forces we assume that the two primaries create a quasi-homogeneous potential, which means that we insert to the inverse square law of gravitation an inverse cube corrective term in order to approximate various phenomena as the radiation pressure of the primaries or the non-sphericity of them. Based on this new consideration we investigate the equilibrium locations of the small body and their parametric dependence, as well as the zero-velocity curves and surfaces for the planar motion, and the evolution of the regions where this motion is permitted when the Jacobian constant varies. [ABSTRACT FROM AUTHOR]

We consider a planar system of five bodies respectively located at $\mathbf{r}_{0}$ , $\mathbf{r}_{1}$ , $\mathbf{r}_{2}$ , $\mathbf{r}_{3}$ and $\mathbf{r}_{4}$ with masses $m_{0} \geq 0$ , $m_{1}= m_{2}= m$ and $m_{3}= m_{4}= \tilde{m}$ moving on a plane. The body of mass $m_{0}$ is considered to be at the center of the configuration which is assumed to be the origin of the coordinate system. The other four bodies, called the primaries, form a rhombus configuration at all time. We assume that the mutual attraction among all the bodies is of the Newtonian type. [ABSTRACT FROM AUTHOR]

We propose a model for modified gravity that meets the conditions of viability. The model has stable constant curvature solution and for an special case contains flat space time solution. The model also leads to matter stability under small perturbations of scalar curvature. We give an example of restrictions on the parameters that give large enough scalaron mass to avoid detectable corrections to the Newton's law. At large curvature the model tends to an effective cosmological constant that leads to the exponential expansion needed for inflation, and at small curvature the model gives almost constant density compatible with current observations of cosmological constant. [ABSTRACT FROM AUTHOR]

We have shown that flat rotation curve in the framework of the generalized Newton's law of gravitation need not require dark matter. The generalized Newton's force gives rise to logarithmic potential energy at large distances from the center of the galaxy. The effect of the gravitomagnetic force is to induce the constant velocity pattern observed in flat rotation curve at very large distances from the center of the galaxy. Dynamical matter arising from moving stars far away from the center of the galaxy is shown to provide a great part to the total mass of the galaxy. [ABSTRACT FROM AUTHOR]

Up to now, the rotational velocities of galaxies are not clearly understood and the experimental Tully Fisher rule, linking the total galactic mass to the fourth power of the velocity, through an acceleration coefficient of about 10 m/s has not found a deep theoretical explanation. Tentative proposals (MOND theory of a modified Newton's law and extraneous dark matter) do not bring a definite clarification. We propose here a new approach to this problem, without exotic matter and using the classical Newton force. But we introduce a new additional universal acceleration, which could represent a universal expansion law valid at the scale level of a galaxy. We show that this hypothesis leads to a good description of the observed variations of the galactic transverse velocity. It can be considered as a consequence of the Scale Expansion Cosmos theory (SEC) introduced by J. Masreliez, but we postulate that the space expansion acceleration universally applies at any scale. We obtain a formal derivation of the Tully Fisher law, linking the constant galactic transverse velocity to its total mass, via the universal minimum acceleration. We derive a good estimate of the TF acceleration coefficient and show that expansion should be proportional to the square root of the local volumic mass density. Our conjecture is in fact a new dynamics principle which could be applied to many other physical problems at different scales. Applying it to the range of the solar planet system confirms the well known Kepler laws, at least as a valid approximation for the order of magnitude of the solar system. [ABSTRACT FROM AUTHOR]

There has been a renewed interest in the recent years in the possibility of deviations from the predictions of Newton's 'inverse-square law' of universal gravitation. One of the reasons for renewing this interest lies in various theoretical attempts to construct a unified elementary particle theory, in which there is a natural prediction of new forces over macroscopic distances. In this paper we study the entropic gravity correction to the gravitational force on the horizon of a black hole whose metric has been modified by a Yukawa term. We find that the gravitational radius of such a black hole is given in-terms of the Lambert function, and the entropic force introduces a extra term that depends on the square of the coupling constant α of the Yukawa potential. In the case alpha equals zero we recover the Newtonian gravitational force on the horizon. In a first effort to obtain a relation between geometry and information, we calculate the Ricci scalar and through entropy we establish a relation to the number of information N where this is given in nats. Finally, we calculate a critical entropy value as well as a critical information number N for which the curvature becomes identically zero which implies that the space becomes flat. [ABSTRACT FROM AUTHOR]

The restricted (equilateral) four-body problem consists of three bodies of masses m, m and m (called primaries) lying in a Lagrangian configuration of the three-body problem i.e., they remain fixed at the apices of an equilateral triangle in a rotating coordinate system. A massless fourth body moves under the Newtonian gravitation law due to the three primaries; as in the restricted three-body problem (R3BP), the fourth mass does not affect the motion of the three primaries. In this paper we explore symmetric periodic orbits of the restricted four-body problem (R4BP) for the case of two equal masses where they satisfy approximately the Routh's critical value. We will classify them in nine families of periodic orbits. We offer an exhaustive study of each family and the stability of each of them. [ABSTRACT FROM AUTHOR]

The far-reaching gravitational force-in the approximation of Newton's law of gravitation-is described by a heuristic model with hypothetical massless particles propagating at the speed of light in vacuum and transferring momentum and energy between physical entities through interactions on a local basis. The model has some similarities with the impact theory presented by Nicolas Fatio de Duillier to the Royal Society in 1690. Objections raised against this idea are dispelled by invoking the Special Theory of Relativity, considering non-local interactions, and replacing the shielding concept by a secular mass increase of massive bodies. Some consequences and applications of the model are discussed. [ABSTRACT FROM AUTHOR]

An impact model of gravity designed to emulate Newton's law of gravitation is applied to particles with relative motions at slow and relativistic speeds. Based on this model, a gravitational interaction mode is then conceived between photons and massive particles. This implies a deflection perpendicular to the propagation direction of a photon twice as large as expected from the mass-energy relation of photons-in accordance with observations and the General Theory of Relativity. The longitudinal interaction is compatible with the energy and momentum conservation principles applied to massless entities, and the results are consistent with the observed Shapiro delay. [ABSTRACT FROM AUTHOR]

We comment the analogies and differences between self and rotational energies for a sphere in Newtonian and General Relativistic theories. Such difference is not trivial. The results have astrophysical importance. [ABSTRACT FROM AUTHOR]

The restricted four-body problem consists of an infinitesimal particle which is moving under the Newtonian gravitational attraction of three finite bodies, m 1 , m 2 and m 3 . The three bodies (called primaries) are moving in circular orbits around their common centre of mass fixed at the origin of the coordinate system. Moreover, according to the solution of Lagrange, these primaries are fixed at the vertices of an equilateral triangle. The fourth body does not affect the motion of the three bodies. In this paper, we deal with the photogravitational version of the problem with Stokes drag acting as a dissipative force. We consider the case where all the primaries are sources of radiation and that two of the bodies, m 2 and m 3 , have equal masses ( m 2 = m 3 = μ ) and equal radiation factors ( q 2 = q 3 = q ) while the dominant primary body m 1 is of mass 1 − 2 μ . We investigate the dynamical behaviour of an infinitesimal mass in the gravitational field of radiating primaries coupled with the Stokes drag effect. It is found that under constant dissipative force, collinear equilibrium points do not exist (numerically and of course analytically) whereas the existence and positions of the non-collinear equilibrium points depend on the parameters values. The linear stability of the non-collinear equilibrium points ( L i , i = 1 , 2 , ... , 8 ) is also studied and it is found that they are all unstable except L 1 , L 7 and L 8 which may be stable for a range of values of μ and various values of radiation factors. Finally, we justify the relevance of the model in astronomy by applying it to a stellar system (Ross 104-Ross775a-Ross775b), for which all the equilibrium points have been seen to be unstable. [ABSTRACT FROM AUTHOR]