The N-body problem

2018 ◽  
Author(s):  
Nathalie Deruelle ◽  
Jean-Philippe Uzan
Keyword(s):  
Chaos Theory ◽  
Scientific Community ◽  
Body Problem ◽  
Equations Of Motion ◽  
60Th Birthday ◽  
Henri Poincaré ◽  
N Body Problem ◽  
Newton’S Laws

This chapter discusses the N-body problem. In 1886, Karl Weierstrass submitted the following question to the scientific community on the occasion of a mathematical competition to mark the 60th birthday of King Oscar II of Sweden. Weierstrass asked that, ‘given a system of arbitrarily many mass points that attract each other according to Newton’s laws, try to find, under the assumption that no two points ever collide, a representation of the coordinates of each point as a series in a variable which is some known function of time and for all of whose values the series converges uniformly’. Henri Poincaré showed that the equations of motion for more than two gravitational bodies are not in general integrable and won the competition. However, the jury awarded the prize to Poincaré not for solving the problem, but for coming up with the first ideas of what later became known as chaos theory.

scholarly journals On the foundations of general relativistic celestial mechanics

2017 ◽  
Vol 32 (26) ◽  
pp. 1730022 ◽  
Author(s):  
Emmanuele Battista ◽  
Giampiero Esposito ◽  
Simone Dell’Agnello
Keyword(s):  
General Relativity ◽  
Solar System ◽  
Celestial Mechanics ◽  
Body Problem ◽  
Equations Of Motion ◽  
Modern Science ◽  
N Body Problem ◽  
General Relativistic ◽  
New Perspective ◽  
Three Body

Towards the end of nineteenth century, Celestial Mechanics provided the most powerful tools to test Newtonian gravity in the solar system and also led to the discovery of chaos in modern science. Nowadays, in light of general relativity, Celestial Mechanics leads to a new perspective on the motion of satellites and planets. The reader is here introduced to the modern formulation of the problem of motion, following what the leaders in the field have been teaching since the nineties, in particular, the use of a global chart for the overall dynamics of N bodies and N local charts describing the internal dynamics of each body. The next logical step studies in detail how to split the N-body problem into two sub-problems concerning the internal and external dynamics, how to achieve the effacement properties that would allow a decoupling of the two sub-problems, how to define external-potential-effacing coordinates and how to generalize the Newtonian multipole and tidal moments. The review paper ends with an assessment of the nonlocal equations of motion obtained within such a framework, a description of the modifications induced by general relativity on the theoretical analysis of the Newtonian three-body problem, and a mention of the potentialities of the analysis of solar-system metric data carried out with the Planetary Ephemeris Program.

Motions of Bodies—Newton’s Laws

2017 ◽  
Author(s):  
Nicholas Manton ◽  
Nicholas Mee
Keyword(s):  
Binary Stars ◽  
Body Problem ◽  
Equations Of Motion ◽  
Three Dimensions ◽  
Planetary Motion ◽  
Harmonic Oscillator Potential ◽  
Least Action ◽  
Alternative Approach ◽  
Principle Of Least Action ◽  
Newton’S Laws

Chapter 2 covers Newtonian dynamics, Newton’s law of gravitation and the motion of mutually gravitating bodies. The principle of least action is used to provide an alternative approach to Newton’s laws. Motion of several bodies is described. By analogy the same results are used to describe the motion of a single body in three dimensions. The equations of motion are solved for a harmonic oscillator potential. The general central potential is considered. The equations are solved for an attractive inverse square law force and shown to agree with Kepler’s laws of planetary motion. The Michell–Cavendish experiment to determine Newton’s gravitational constant is described. The centre of mass is defined and the motion of composite bodies described. The Kepler 2-body problem is solved and applied to binary stars. The positions of the five Lagrangian points are calculated. Energy conservation in mechanical systems is discussed, and friction and dissipation are considered.

scholarly journals The Classical N-body Problem in the Context of Curved Space

2017 ◽  
Vol 69 (4) ◽  
pp. 790-806 ◽  
Author(s):  
Florin Diacu
Keyword(s):  
Gaussian Curvature ◽  
Body Problem ◽  
Equations Of Motion ◽  
Classical Case ◽  
Natural Generalization ◽  
First Integrals ◽  
Euclidean Case ◽  
Constant Gaussian Curvature ◽  
Curved Space ◽  
N Body Problem

AbstractWe provide the differential equations that generalize the Newtonian N-body problem of celestial mechanics to spaces of constant Gaussian curvature κ, for all κ ∊ ℝ. In previous studies, the equations of motion made sense only for κ ≠ 0. The system derived here does more than just include the Euclidean case in the limit κ → 0; it recovers the classical equations for κ = 0. This new expression of the laws of motion allows the study of the N-body problem in the context of constant curvature spaces and thus oòers a natural generalization of the Newtonian equations that includes the classical case. We end the paper with remarks about the bifurcations of the first integrals.

The post-Newtonian approximation

2018 ◽  
Author(s):  
Nathalie Deruelle ◽  
Jean-Philippe Uzan
Keyword(s):  
General Relativity ◽  
Gravitational Field ◽  
Body Problem ◽  
Equations Of Motion ◽  
Two Body Problem ◽  
Newtonian Approximation ◽  
Actual Motion ◽  
Law Of Attraction ◽  
Universal Law ◽  
Two Bodies

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.

The two-body problem: an effective-one-body approach

2018 ◽  
Author(s):  
Nathalie Deruelle ◽  
Jean-Philippe Uzan
Keyword(s):  
Body Problem ◽  
Equations Of Motion ◽  
Reaction Force ◽  
Kerr Black Hole ◽  
Radiation Reaction ◽  
Spherically Symmetric ◽  
Geodesic Motion ◽  
Two Body Problem ◽  
Symmetric Spacetime ◽  
Radiation Reaction Force

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.

scholarly journals On frequency estimations in phase fitted variational integrators for the general N-body problem

PAMM ◽  
2013 ◽  
Vol 13 (1) ◽  
pp. 33-34
Author(s):  
Odysseas Kosmas ◽  
Sigrid Leyendecker
Keyword(s):  
Body Problem ◽  
Variational Integrators ◽  
N Body Problem
Sign in / Sign up

Export Citation Format

Share Document