Qualitative Aspects of theN-Body Problem for Approximately Relativistic Equations of Motion

1965 ◽  
Vol 140 (4B) ◽  
pp. B1162-B1173 ◽  
Author(s):  
Richard B. Hoffman ◽  
Peter Havas
1974 ◽  
Vol 64 ◽  
pp. 102-102
Author(s):  
Arnold Rosenblum

A new general covariant approach to the general relativistic equations of motion is presented. It is stressed that our present understanding of the development of binaries due to general relativistic effects and of the power emitted by these systems in the form of gravitational radiation is highly unsatisfactory.


2003 ◽  
Vol 58 (1) ◽  
pp. 13-22 ◽  
Author(s):  
Lucas F. Wanex

The relativistic equations of motion for the restricted three-body problem are derived in the first post-Newtonian approximation. These equations are integrated numerically for seven different trajectories in the earth-moon orbital system. Four of the trajectories are determined to be chaotic and three are not chaotic. Each post-Newtonian trajectory is compared to its Newtonian counterpart. It is found that the difference between Newtonian and post-Newtonian trajectories for the restricted three-body problem is greater for chaotic trajectories than it is for trajectories that are not chaotic. Finally, the possibility of using this Chaotic Amplification Effect as a novel test of general relativity is discussed.


Author(s):  
Nathalie Deruelle ◽  
Jean-Philippe Uzan

This chapter embarks on a study of the two-body problem in general relativity. In other words, it seeks to describe the motion of two compact, self-gravitating bodies which are far-separated and moving slowly. It limits the discussion to corrections proportional to v2 ~ m/R, the so-called post-Newtonian or 1PN corrections to Newton’s universal law of attraction. The chapter first examines the gravitational field, that is, the metric, created by the two bodies. It then derives the equations of motion, and finally the actual motion, that is, the post-Keplerian trajectories, which generalize the post-Keplerian geodesics obtained earlier in the chapter.


Author(s):  
Nathalie Deruelle ◽  
Jean-Philippe Uzan

This chapter presents the basics of the ‘effective-one-body’ approach to the two-body problem in general relativity. It also shows that the 2PN equations of motion can be mapped. This can be done by means of an appropriate canonical transformation, to a geodesic motion in a static, spherically symmetric spacetime, thus considerably simplifying the dynamics. Then, including the 2.5PN radiation reaction force in the (resummed) equations of motion, this chapter provides the waveform during the inspiral, merger, and ringdown phases of the coalescence of two non-spinning black holes into a final Kerr black hole. The chapter also comments on the current developments of this approach, which is instrumental in building the libraries of waveform templates that are needed to analyze the data collected by the current gravitational wave detectors.


Author(s):  
S. E. Abd El-Bar

Under the influence of some different perturbations, we study the stability of collinear equilibrium points of the Restricted Three Body Problem. More precisely, the perturbations due to the triaxiality of the bigger primary and the oblateness of the smaller primary, in addition to the relativistic effects, are considered. Moreover, the total potential and the mean motion of the problem are obtained. The equations of motion are derived and linearized around the collinear points. For studying the stability of these points, the characteristic equation and its partial derivatives are derived. Two real and two imaginary roots of the characteristic equation are deduced from the plotted figures throughout the manuscript. In addition, the instability of the collinear points is stressed. Finally, we compute some selected roots corresponding to the eigenvalues which are based on some selected values of the perturbing parameters in the Tables 1, 2.


2008 ◽  
Vol 74 (1) ◽  
pp. 111-118
Author(s):  
FEN-CE CHEN

AbstractThe acceleration of ions by multiple laser pulses and their spontaneously generated electric and magnetic fields is investigated by using an analytical model for the latter. The relativistic equations of motion of test charged particles are solved numerically. It is found that the self-generated axial electric field plays an important role in the acceleration, and the energy of heavy test ions can reach several gigaelectronvolts.


1974 ◽  
Vol 62 ◽  
pp. 63-69 ◽  
Author(s):  
G. A. Chebotarev ◽  
N. A. Belyaev ◽  
R. P. Eremenko

In this paper the orbital evolution of Trojan asteroids are studied by integrating numerically the equations of motion over the interval 1660–2060, perturbations from Venus to Pluto being taken into account. The comparison of the actual motion of Trojans in the solar system with the theory based on the restricted three-body problem are given.


Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 553 ◽  
Author(s):  
Essam R. El-Zahar ◽  
Abdelhalim Ebaid ◽  
Abdulrahman F. Aljohani ◽  
José Tenreiro Machado ◽  
Dumitru Baleanu

This paper re-analyzes the falling body problem in three dimensions, taking into account the effect of the Earth’s rotation (ER). Accordingly, the analytic solution of the three-dimensional model is obtained. Since the ER is quite slow, the three coupled differential equations of motion are usually approximated by neglecting all high order terms. Furthermore, the theoretical aspects describing the nature of the falling point in the rotating frame and the original inertial frame are proved. The theoretical and numerical results are illustrated and discussed.


1971 ◽  
Vol 10 ◽  
pp. 133-150 ◽  
Author(s):  
D. G. Bettis ◽  
V. Szebehely

AbstractOne of the main difficulties encountered in the numerical integration of the gravitational n-body problem is associated with close approaches. The singularities of the differential equations of motion result in losses of accuracy and in considerable increase in computer time when any of the distances between the participating bodies decreases below a certain value. This value is larger than the distance when tidal effects become important, consequently, numerical problems are encountered before the physical picture is changed. Elimination of these singularities by transformations is known as the process of regularization. This paper discusses such transformations and describes in considerable detail the numerical approaches to more accurate and faster integration. The basic ideas of smoothing and regularization are explained and applications are given.


2015 ◽  
Vol 33 (2) ◽  
pp. 307-313 ◽  
Author(s):  
M. A. Pocsai ◽  
S. Varró ◽  
I. F. Barna

AbstractAn effective theory of laser–plasma-based particle acceleration is presented. Here we treated the plasma as a continuous medium with an index of refraction nm in which a single electron propagates. Because of the simplicity of this model, we did not perform particle-in-cell (PIC) simulations in order to study the properties of the electron acceleration. We studied the properties of the electron motion due to the Lorentz force and the relativistic equations of motion were numerically solved and analyzed. We compared our results with PIC simulations and experimental data.


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