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## Stability in the Full Two-Body Problem

### Celestial Mechanics and Dynamical Astronomy (2002-05-01) 83: 155-169 , May 01, 2002

Stability conditions are established in the problem of two gravitationally interacting rigid bodies, designated here as the full two-body problem. The stability conditions are derived using basic principles from the *N*-body problem which can be carried over to the full two-body problem. Sufficient conditions for Hill stability and instability, and for stability against impact are derived. The analysis is applicable to binary small-body systems such as have been found recently for asteroids and Kuiper belt objects.

## Stability of resonant planetary orbits in binary stars

### Journal of Astrophysics and Astronomy (1989-12-01) 10: 347-365 , December 01, 1989

This paper contains a numerical study of the stability of resonant orbits in a planetary system consisting of two planets, moving under the gravitational attraction of a binary star. Its results are expected to provide us with useful information about real planetary systems and, at the same time, about periodic motions in the general four-body problem (G4) because the above system is a special case of G4 where two bodies have much larger masses than the masses of the other two (planets). The numerical results show that the main mechanism which generates instability is the destruction of the Jacobi integrals of the massless planets when their masses become nonzero and that resonances in the motion of planets do not imply, in general, instability. Considerable intervals of stable resonant orbits have been found. The above quantitative results are in agreement with the existing qualitative predictions

## Effect of perturbed potentials on the non-linear stability of libration pointL 4 in the restricted problem

### Celestial Mechanics and Dynamical Astronomy (1994-08-01) 59: 345-374 , August 01, 1994

The non-linear stability of the libration point*L*_{4} in the restricted problem has been studied when there are perturbations in the potentials between the bodies. It is seen that the point*L*_{4} is stable for all mass ratios in the range of linear stability except for three mass ratios depending upon the perturbing functions. The theory is applied to the following four cases:
(i)

There are no perturbations in the potentials (classical problem).

(ii)Only the bigger primary is an oblate spheroid whose axis of symmetry is perpendicular to the plane of relative motion (circular) of the primaries.

(iii)Both the primaries are oblate spheroids whose axes of symmetry are perpendicular to the plane of relative motion (circular) of the primaries.

(iv)The primaries are spherical in shape and the bigger is a source of radiation.

## Properties of Optical Phase-Locked Loop Based on Four Wave Mixing in Semiconductor Laser Amplifiers

### International Journal of Infrared and Millimeter Waves (1998-12-01) 19: 1721-1734 , December 01, 1998

In this paper, the properties of the optical phase-locked loop(PLL) based on the four-wave mixing in the semiconductor laser amplifiers (SLAs) are discussed. The components that achieve the function of detecting the bit phase of the input optical signal are concerned and discussed in detail together as a function module named as the optical bit phase detector referred to the general electronic PLL. Therefore, most of the properties of the optical PLL can be analyzed by applying the general phase-locked theory. Here the stability of the optical PLL is discussed. It's shown that the variance of input signal power in the practical application will cause optical PLL system unstable because of its long loop delay. The influence on the output phase jitter of the optical PLL is also investigated.

## Stability of Hamiltonian Systems with Three Degrees of Freedom and the Three Body-Problem

### Celestial Mechanics and Dynamical Astronomy (2006-03-01) 94: 249-269 , March 01, 2006

Results are obtained about formal stability and instability of Hamiltonian systems with three degrees of freedom, two equal frequencies and the matrix of the linear part is not diagonalizable, in terms of the coefficients of the development in Taylor series of the Hamiltonian of the system. The results are applied to the study of stability of the Lagrangian solutions of the Three Body-Problem in the case in which the center of mass is over the curve ρ_{*}, on the border of the region of linear stability of Routh. The curve ρ_{*} is divided symmetrically in three arcs in such a way that if the center of mass of the three particles lies on the central arc, the Lagrangian solution is unstable in the sense of Liapunov (in finite order), while if the center of mass determines one point that lies on one of the other two arcs of ρ_{*}, then the Lagrangian solution is formally stable.

## On the Stability of Planar Oscillations and Rotations of a Satellite in a Circular Orbit

### Celestial Mechanics and Dynamical Astronomy (2003-01-01) 85: 51-66 , January 01, 2003

We deal with the stability problem of planar periodic motions of a satellite about its center of mass. The satellite is regarded a dynamically symmetric rigid body whose center of mass moves in a circular orbit.

By using the method of normal forms and KAM theory we study the orbital stability of planar oscillations and rotations of the satellite in detail. In two special cases we investigate the orbital stability analytically by introducing a small parameter. In the general case, numerical calculations of Hamiltonian normal form are necessary.

## Effective Hamiltonian for the D'Alembert Planetary Model Near a Spin/Orbit Resonance

### Celestial Mechanics and Dynamical Astronomy (2002-05-01) 83: 223-237 , May 01, 2002

The D'Alembert model for the spin/orbit problem in celestial mechanics is considered. Using a Hamiltonian formalism, it is shown that in a small neighborhood of a *p*:*q* spin/orbit resonance with (*p*,*q*) different from (1,1) and (2,1) the 'effective' D'Alembert Hamiltonian is a completely integrable system with phase space foliated by maximal invariant curves; instead, in a small neighborhood of a *p*:*q* spin/orbit resonance with (*p*,*q*) equal to (1,1) or (2,1) the 'effective' D'Alembert Hamiltonian has a phase portrait similar to that of the standard pendulum (elliptic and hyperbolic equilibria, separatrices, invariant curves of different homotopy). A fast averaging with respect to the 'mean anomaly' is also performed (by means of Nekhoroshev techniques) showing that, up to exponentially small terms, the resonant D'Alembert Hamiltonian is described by a two-degrees-of-freedom, properly degenerate Hamiltonian having the lowest order terms corresponding to the 'effective' Hamiltonian mentioned above.

## Research for 3mm Band IF-Switch Radiometer

### International Journal of Infrared and Millimeter Waves (2001-06-01) 22: 887-893 , June 01, 2001

A 3mm band IF-switch Radiometer was proposed in this paper. On the basis of discussing the theory and circuit structure features, the performance of experimental prototype was presented. The research and experiment result shows that not only its stability is better than that of total power radiometer, but also its cost and size are just similar as the latter. In the absence of 3mm band switch it can surely meet the needs of practical use.

## Sur l'instabilite de certaines positions d'equilibre relatif dans le probleme des n corps

### Celestial Mechanics and Dynamical Astronomy (1990-09-01) 49: 219-231 , September 01, 1990

### Resumé

On démontre dans cet article l'instabilité, pour tout *n* ⩾ 4, des configurations d'équilibre relatif dans le problème des *n* corps, oú les *n* corps soumises aux attractions newtonniennes mutuelles se trouvent aux sommets d'un polygone régulier de *n* cotés. La preuve consiste à montrer que les équations aux variations, projetées sur le plan *P* des *n* corps, possèdent au moins deux exposants caractéristiques complexes connugués dont la parr'e réelle est strictement positive; alors que ces equations projetées sur un axe orthogonal à *P* possèdent des solutions ayant des termes séculaires.

## Structure and stability of rotating fluid disks around massive objects. I. Newtonian formulation

### Journal of Astrophysics and Astronomy (1981-12-01) 2: 421-437 , December 01, 1981

In this paper we have presented a very general class of solutions for rotating fluid disks around massive objects (neglecting the self gravitation of the disk) with density as a function of the radial coordinate only and pressure being nonzero. Having considered a number of cases with different density and velocity distributions, we have analysed the stability of such disks under both radial and axisymmetric perturbations. For a perfect gas disk with γ*=* 5/3 the disk is stable with frequency (MG/r^{3})^{1/2} for purely radial pulsation with expanding and contracting boundary. In the case of axisymmetric perturbation the critical γ_{c} for neutral stability is found to be much less than 4/3 indicating that such disks are mostly stable under such perturbations.