On the Metric Stability and the Nekhoroshev Estimate of the Velocity of Arnold Diffusion in a Special Case of the Three-body Problem

2021 ◽  
Vol 26 (4) ◽  
pp. 321-330
Author(s):  
Anatoly P. Markeev
Keyword(s):  
Body Problem ◽  
Three Body Problem ◽  
Arnold Diffusion ◽  
Three Body ◽  
Special Case

scholarly journals Nekhoroshev-Stability ofL4andL5in the Spatial Restricted Problem

1999 ◽  
Vol 172 ◽  
pp. 445-446 ◽  
Author(s):  
Giancarlo Benettin ◽  
Francesco Fassò ◽  
Massimiliano Guzzo
Keyword(s):  
Restricted Problem ◽  
Body Problem ◽  
Critical Mass ◽  
Kam Theory ◽  
Birkhoff Normal Form ◽  
Three Body Problem ◽  
Arnold Diffusion ◽  
Convexity Assumption ◽  
Three Body ◽  
Nekhoroshev Stability

The Lagrangian equilateral pointsL4andL5of the restricted circular three-body problem are elliptic for all values of the reduced massμbelow Routh’s critical massμR≈ .0385. In the spatial case, because of the possibility of Arnold diffusion, KAM theory does not provide Lyapunov-stability. Nevertheless, one can consider the so-called ‘Nekhoroshev-stability’: denoting byda convenient distance from the equilibrium point, one asks whetherfor any small єe > 0, with positiveaandb. Until recently this problem, as more generally the problem of Nekhoroshev-stability of elliptic equilibria of Hamiltonian systems, was studied only under some arithmetic conditions on the frequencies, and thus onμ(see e.g .Giorgilli, 1989). Our aim was instead considering all values ofμup toμR. As a matter of fact, Nekhoroshev-stability of elliptic equilibria, without any arithmetic assumption on the frequencies, was proved recently under the hypothesis that the fourth order Birkhoff normal form of the Hamiltonian exists and satisfies a ‘quasi-convexity’ assumption (Fassòet al, 1998; Guzzoet al, 1998; Niedermann, 1998).

scholarly journals Stability of the Sitnikov's circular restricted three body problem when the primaries are oblate spheroid

BIBECHANA ◽  
2014 ◽  
Vol 11 ◽  
pp. 149-156
Author(s):  
RR Thapa
Keyword(s):  
Body Problem ◽  
Equilibrium Points ◽  
Oblate Spheroid ◽  
Restricted Three Body Problem ◽  
Three Body Problem ◽  
Third Body ◽  
Independent Variable ◽  
The Stability ◽  
Three Body ◽  
Special Case

The Sitnikov's problem is a special case of restricted three body problem if the primaries are of equal masses (m1 = m2 = 1/2) moving in circular orbits under Newtonian force of attraction and the third body of mass m3 moves along the line perpendicular to plane of motion of primaries. Here oblate spheroid primaries are taken. The solution of the Sitnikov's circular restricted three body problem has been checked when the primaries are oblate spheroid. We observed that solution is depended on oblate parameter A of the primaries and independent variable τ = ηt. For this the stability of non-trivial solutions with the characteristic equation is studied. The general equation of motion of the infinitesimal mass under mutual gravitational field of two oblate primaries are seen at equilibrium points. Then the stability of infinitesimal third body m3 has been calculated. DOI: http://dx.doi.org/10.3126/bibechana.v11i0.10395 BIBECHANA 11(1) (2014) 149-156

DIFFERENT TYPES OF ATTRACTORS IN THE THREE BODY PROBLEM PERTURBED BY DISSIPATIVE TERMS

2011 ◽  
Vol 21 (08) ◽  
pp. 2195-2209 ◽  
Author(s):  
JOHN D. HADJIDEMETRIOU ◽  
GEORGE VOYATZIS
Keyword(s):  
Hamiltonian System ◽  
Degrees Of Freedom ◽  
Body Problem ◽  
Large Mass ◽  
Three Body Problem ◽  
Resonant Orbits ◽  
Different Types ◽  
Families Of Periodic Orbits ◽  
Three Body ◽  
Special Case

We study the evolution of a conservative dynamical system with three degrees of freedom, where small nonconservative terms are added. The conservative part is a Hamiltonian system, describing the motion of a planetary system consisting of a star, with a large mass, and of two planets, with small but not negligible masses, that interact gravitationally. This is a special case of the three body problem, which is nonintegrable. We show that the evolution of the system follows the topology of the conservative part. This topology is critically determined by the families of periodic orbits and their stability. The evolution of the complete system follows the families of the conservative part and is finally trapped in the resonant orbits of the Hamiltonian system, in different types of attractors: chaotic attractors, limit cycles or fixed points.

scholarly journals The special case of the three body problem, when gravitational potential is given as the Kislik potential

2014 ◽  
Vol 9 (S310) ◽  
pp. 45-48
Author(s):  
A. Shuvalova ◽  
T. Salnikova
Keyword(s):  
Body Problem ◽  
Relative Equilibria ◽  
Point Mass ◽  
Restricted Three Body Problem ◽  
The Moon ◽  
Three Body Problem ◽  
The Earth ◽  
The Difference ◽  
Three Body ◽  
Special Case

AbstractIn this paper we consider the special case of the planar circular restricted three-body problem by the example of the problem of the Earth, the Moon and a point mass, where the gravitational potentials of the Earth and the Moon are given as the Kislik potential. The Kislik potential takes into account the flattening of a celestial body on the poles. We find the relative equilibria solutions for a point mass and analyze their stability. We describe the difference between the obtained points and the classical solution of the three-body problem.

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