scholarly journals The zonal satellite problem. III Symmetries

2002 ◽  
pp. 1-8
Author(s):  
V. Mioc
Keyword(s):  
Force Field ◽  
Body Problem ◽  
Blow Up ◽  
Polar Coordinates ◽  
Two Body Problem ◽  
Satellite Problem ◽  
Group Of Symmetries

The two-body problem associated with a force field described by a potential of the form U =Sum(k=1,n) ak/rk (r = distance between particles, ak = real parameters) is resumed from the only standpoint of symmetries. Such symmetries, expressed in Hamiltonian coordinates, or in standard polar coordinates, are recovered for McGehee-type coordinates of both collision-blow-up and infinity-blow-up kind. They form diffeomorphic commutative groups endowed with a Boolean structure. Expressed in Levi-Civita?s coordinates, the problem exhibits a larger group of symmetries, also commutative and presenting a Boolean structure.

scholarly journals Symmetries in central-force problems

2003 ◽  
pp. 47-52 ◽  
Author(s):  
V. Mioc ◽  
M. Barbosu
Keyword(s):  
Vector Fields ◽  
Body Problem ◽  
Blow Up ◽  
Central Force ◽  
Polar Coordinates ◽  
Global Flow ◽  
Two Body Problem ◽  
Concrete Problem ◽  
Central Force Problems ◽  
Force Problem

The two-body problem in central fields (reducible to a central-force problem) models a lot of concrete astronomical situations. The corresponding vector fields (in Cartesian and polar coordinates, extended via collision-blow-up and infinity-blow-up transformations) exhibit nice symmetries that form eight-element Abelian groups endowed with an idempotent structure. All these groups are isomorphic, which is not a trivial result, given the different structures of the corresponding phase spaces. Each of these groups contains seven four-element subgroups isomorphic to Klein?s group. These symmetries are of much help in understanding various characteristics of the global flow of the general problem or of a concrete problem at hand, and are essential in searching for periodic orbits.

scholarly journals The zonal satellite problem - I: Near-collision flow

1998 ◽  
pp. 31-36 ◽  
Author(s):  
V. Mioc ◽  
M. Stavinschi
Keyword(s):  
Angular Momentum ◽  
Blow Up ◽  
Rotational Symmetry ◽  
Local Flow ◽  
Physical Motion ◽  
Two Body Problem ◽  
Collision Singularity ◽  
Satellite Problem ◽  
Collision Manifold ◽  
Momentum Integral

The force field described by a potential function of the form U = ?n k=1 ak/rk (r = distance between particles, ak = real parameters) models various concrete situations belonging to astronomy, physics, mechanics, astrodynamics, etc. The two-body problem is being tackled in such a field. The motion equations and the first integrals of energy and angular momentum are established. The McGehee-type coordinates are used to blow up the collision singularity and to paste the resulting manifold on the phase space. The flow on the collision manifold is depicted. Then, using the rotational symmetry of the problem and the angular momentum integral, the local flow near collision is described and interpreted in terms of physical motion.

scholarly journals Collision dynamics in Hénon-Heiles’ two-body problem

2003 ◽  
pp. 43-46 ◽  
Author(s):  
V. Mioc ◽  
M. Barbosu
Keyword(s):  
Body Problem ◽  
Blow Up ◽  
Collision Dynamics ◽  
Two Body Problem ◽  
Collision Singularity ◽  
Collision Manifold ◽  
Special Case

We tackle the two-body problem associated to H?non-Heiles? potential in the special case of the collision singularity. Using McGehee-type transformations of the second kind, we blow up the singularity and replace it by the collision manifold Mc pasted on the phase spece. We fully describe the flow on Mc. This flow is similar to analogous flows met in post-Newtonian two-body problems.

scholarly journals Geometric analysis of a two-body problem with quick loss of mass

2021 ◽  
Author(s):  
Zhouqian Miao ◽  
Nikola Popović ◽  
Thomas Zacharis
Keyword(s):  
Singular Perturbation ◽  
Vector Fields ◽  
Body Problem ◽  
Blow Up ◽  
Geometric Approach ◽  
Resonance Phenomenon ◽  
Singular Perturbation Theory ◽  
Geometric Singular Perturbation Theory ◽  
Two Body Problem ◽  
Loss Of Mass

AbstractWe consider a two-body problem with quick loss of mass which was formulated by Verhulst (Verhulst in J Inst Math Appl 18: 87–98, 1976). The corresponding dynamical system is singularly perturbed due to the presence of a small parameter in the governing equations which corresponds to the reciprocal of the initial rate of loss of mass, resulting in a boundary layer in the asymptotics. Here, we showcase a geometric approach which allows us to derive asymptotic expansions for the solutions of that problem via a combination of geometric singular perturbation theory (Fenichel in J Differ Equ 31: 53–98, 1979) and the desingularization technique known as “blow-up” (Dumortier, in: Bifurcations and Periodic Orbits of Vector Fields, Springer, Dordrecht, 1993). In particular, we justify the unexpected dependence of those expansions on fractional powers of the singular perturbation parameter; moreover, we show that the occurrence of logarithmic (“switchback”) terms therein is due to a resonance phenomenon that arises in one of the coordinate charts after blow-up.

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.

A Fitted Runge-Kutta-Nyström Method with Six Stages for the Integration of the Two-Body Problem

2011 ◽  
Author(s):  
A. A. Kosti ◽  
Z. A. Anastassi ◽  
T. E. Simos ◽  
Theodore E. Simos ◽  
George Psihoyios ◽  
...  
Keyword(s):  
Body Problem ◽  
Two Body Problem ◽  
Runge Kutta
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