scholarly journals Nekhoroshev Stability in Quasi-Integrable Degenerate Hamiltonian Systems

1999 ◽  
Vol 172 ◽  
pp. 443-444
Author(s):  
Massimiliano Guzzo
Keyword(s):  
Degenerate System ◽  
Escape Time ◽  
Kam Theorem ◽  
Mass Ratio ◽  
Three Body Problem ◽  
Arbitrary Potential ◽  
Hamiltonian Perturbation Theory ◽  
Three Body ◽  
Nekhoroshev Stability ◽  
Degenerate Hamiltonian Systems

Many classical problems of Mechanics can be studied regarding them as perturbations of integrable systems; this is the case of the fast rotations of the rigid body in an arbitrary potential, the restricted three body problem with small values of the mass-ratio, and others. However, the application of the classical results of Hamiltonian Perturbation Theory to these systems encounters difficulties due to the presence of the so-called ‘degeneracy’. More precisely, the Hamiltonian of a quasi-integrable degenerate system looks likewhere (I, φ) є U × Tn, U ⊆ Rn, are action-angle type coordinates, while the degeneracy of the system manifests itself with the presence of the ‘degenerate’ variables (p, q) є B ⊆ R2m. The KAM theorem has been applied under quite general assumptions to degenerate Hamiltonians (Arnold, 1963), while the Nekhoroshev theorem (Nekhoroshev, 1977) provides, if h is convex, the following bounds: there exist positive ε0, a0, t0 such that if ε < ε0 then if where Te is the escape time of the solution from the domain of (1). An escape is possible because the motion of the degenerate variables can be bounded in principle only by , and so over the time they can experience large variations. Therefore, there is the problem of individuating which assumptions on the perturbation and on the initial data allow to control the motion of the degenerate variables over long times.

scholarly journals Progression of bifurcated family f type periodic orbits in the circular restricted three-body problem

2017 ◽  
Vol 5 (2) ◽  
pp. 69
Author(s):  
Nishanth Pushparaj ◽  
Ram Krishan Sharma
Keyword(s):  
Solar Radiation ◽  
Periodic Orbits ◽  
Radiation Pressure ◽  
Body Problem ◽  
Solar Radiation Pressure ◽  
Restricted Three Body Problem ◽  
Mass Ratio ◽  
Three Body Problem ◽  
Semi Major Axis ◽  
Three Body

Progression of f-type family of periodic orbits, their nature, stability and location nearer the smaller primary for different mass ratios in the framework of circular restricted three-body problem is studied using Poincaré surfaces of section. The orbits around the smaller primary are found to decrease in size with increase in Jacobian Constant C, and move very close towards the smaller primary. The orbit bifurcates into two orbits with the increase in C to 4.2. The two orbits that appear for this value of C belong to two adjacent separate families: one as direct orbit belonging to family g of periodic orbits and other one as retrograde orbit belonging to family f of periodic orbits. This bifurcation is interesting. These orbits increase in size with increase in mass ratio. The elliptic orbits found within the mass ratio 0 < µ ≤ 0.1 have eccentricity less than 0.2 and the orbits found above the mass ratio µ > 0.1 are elliptical orbits with eccentricity above 0.2. Deviations in the parameters: eccentricity, semi-major axis and time period of these orbits with solar radiation pressure q are computed in the frame work of photogravitational restricted Three-body problem in addition to the restricted three-body problem. These parameters are found to decrease with increase in the solar radiation pressure.

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 The structure of the co-orbital stable regions as a function of the mass ratio

2020 ◽  
Vol 496 (3) ◽  
pp. 3700-3707
Author(s):  
L Liberato ◽  
O C Winter
Keyword(s):  
Angular Distance ◽  
Stable Region ◽  
Polynomial Functions ◽  
Mass Ratio ◽  
Three Body Problem ◽  
System Parameters ◽  
Wide Range ◽  
Value Range ◽  
Three Body ◽  
Orbital Region

ABSTRACT Although the search for extrasolar co-orbital bodies has not had success so far, it is believed that they must be as common as they are in the Solar system. Co-orbital systems have been widely studied, and there are several works on stability and even on formation. However, for the size and location of the stable regions, authors usually describe their results but do not provide a way to find them without numerical simulations, and, in most cases, the mass ratio value range is small. In this work, we study the structure of co-orbital stable regions for a wide range of mass ratio systems and build empirical equations to describe them. It allows estimating the size and location of co-orbital stable regions from a few system parameters. Thousands of massless particles were distributed in the co-orbital region of a massive secondary body and numerically simulated for a wide range of mass ratios (μ) adopting the planar circular restricted three-body problem. The results show that the upper limit of horseshoe regions is between 9.539 × 10−4 &lt; μ &lt; 1.192 × 10−3, which corresponds to a minimum angular distance from the secondary body to the separatrix of between 27.239º and 27.802º. We also found that the limit to existence of stability in the co-orbital region is about μ = 2.3313 × 10−2, much smaller than the value predicted by the linear theory. Polynomial functions to describe the stable region parameters were found, and they represent estimates of the angular and radial widths of the co-orbital stable regions for any system with 9.547 × 10−5 ≤ μ ≤ 2.331 × 10−2.

scholarly journals A Global Analysis of The Generalized Sitnikov Problem

1999 ◽  
Vol 172 ◽  
pp. 291-302
Author(s):  
Steven R. Chesley
Keyword(s):  
Angular Momentum ◽  
Global Analysis ◽  
Mass Ratio ◽  
Three Body Problem ◽  
Bounded Motion ◽  
Type Symmetry ◽  
The Stability ◽  
Two Parameters ◽  
Three Body ◽  
Area Preserving

AbstractThe isosceles three-body problem with Sitnikov-type symmetry has been reduced to a two-dimensional area-preserving Poincaré map depending on two parameters: the mass ratio, and the total angular momentum. The entire parameter space is explored, contrasting new results with ones obtained previously in the planar (zero angular momentum) case. The region of allowable motion is divided into subregions according to a symbolic dynamics representation. This enables a geometric description of the system based on the intersection of the images of the subregions with the preimages. The paper also describes the regions of allowable motion and bounded motion, and discusses the stability of the dominant periodic orbit.

scholarly journals Retrograde Satellites in the Circular Plane Restricted Three-Body Problem

1974 ◽  
Vol 62 ◽  
pp. 129-129
Author(s):  
D. Benest
Keyword(s):  
Body Problem ◽  
Restricted Three Body Problem ◽  
Mass Ratio ◽  
Three Body Problem ◽  
Three Body ◽  
Circular Plane

Characteristics and stability of simple-periodic retrograde satellites of the lighter body are presented for Hill's case and for all values of the mass ratio m2/(m1+m2) between 0 and 0.5.

The role of the mass ratio in ballistic capture

2020 ◽  
Vol 498 (1) ◽  
pp. 1515-1529
Author(s):  
Zong-Fu Luo
Keyword(s):  
Initial Conditions ◽  
Massless Particle ◽  
Mass Ratio ◽  
Three Body Problem ◽  
Ballistic Capture ◽  
Jacobi Constant ◽  
Celestial Bodies ◽  
Three Body ◽  
Gravity System

ABSTRACT A massless particle can be naturally captured by a celestial body with the aid of a third body. In this work, the influence of the mass ratio on ballistic capture is investigated in the planar circular restricted three-body problem (CR3BP) model. Four typical dynamical environments with decreasing mass ratios, that is, the Pluto–Charon, Earth–Moon, Sun–Jupiter, and Saturn–Titan systems, are considered. A generalized method is introduced to derive ballistic capture orbits by starting from a set of initial conditions and integrating backward in time. Particular attention is paid to the backward escape orbits, following which a test particle can be temporarily trapped by a three-body gravity system, although the particle will eventually deviate away from the system. This approach is applied to the four candidate systems with a series of Jacobi constant levels to survey and compare the capture probability (quantitatively) and capture capability (qualitatively) when the mass ratio varies. Capture mechanisms inducing favourable ballistic capture are discussed. Moreover, the possibility and stability of capture by secondary celestial bodies are analysed. The obtained results may be useful in explaining the capture phenomena of minor bodies or in designing mission trajectories for interplanetary probes.

Perturbed Location of L1 point in the photogravitational relativistic restricted three-body problem (R3BP) with oblate primaries

2015 ◽  
Vol 93 (3) ◽  
pp. 300-311 ◽  
Author(s):  
S.E. Abd El-Bar ◽  
F.A. Abd El-Salam ◽  
M. Rassem
Keyword(s):  
Body Problem ◽  
Restricted Three Body Problem ◽  
Mass Ratio ◽  
Three Body Problem ◽  
Series Form ◽  
Analytical Result ◽  
Three Body

The restricted three-body problem is studied in the post-Newtonian framework. The primaries are assumed oblate radiant sources. The perturbed location of the L1 point is computed, and a series form of the location of this point is obtained as a new analytical result. To introduce a semianalytical view, a Mathematica 9 program is constructed so as to draw the location of L1 versus the mass ratio μ ∈ (0, 0.5) taking into account one or more of the considered perturbations. All the obtained illustrations are analyzed.

scholarly journals On stability of triangular points of the restricted relativistic elliptic three-body problem with triaxial and oblate primaries

2017 ◽  
pp. 47-52
Author(s):  
K. Zahra ◽  
Z. Awad ◽  
H.R. Dwidar ◽  
M. Radwan
Keyword(s):  
Linear Stability ◽  
Body Problem ◽  
Critical Mass ◽  
Combined Effects ◽  
Mass Ratio ◽  
Three Body Problem ◽  
Triangular Points ◽  
The Stability ◽  
Three Body ◽  
Ratio Range

This paper investigates the location and linear stability of triangular points under combined effects of perturbations: triaxialty of a massive primary, oblateness of a less massive one, and relativistic corrections. The primaries in this system are assumed to move in elliptical orbits around their common barycenter. It is found that the locations of the triangular points are affected by the involved perturbations. The stability of orbits near these points is also examined. We observed that these points are stable for the mass ratio, ?, range 0 < ? < ?c, where ?c is the critical mass ratio, and unstable for the range ?c ? ? ? 0.5.

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