Linear Stability of the Lagrangian Triangle Solutions for Quasihomogeneous Potentials

Abstract

In this paper, we study the linear stability of the relative equilibria for homogeneous and quasihomogeneous potentials. First, in the case the potential is a homogeneous function of degree −a, we find that any relative equilibrium of the n-body problem with a > 2 is spectrally unstable. We also find a similar condition in the quasihomogeneous case. Then we consider the case of three bodies and we study the stability of the equilateral triangle relative equilibria. In the case of homogeneous potentials we recover the classical result obtained by Routh in a simpler way. In the case of quasihomogeneous potentials we find a generalization of Routh inequality and we show that, for certain values of the masses, the stability of the relative equilibria depends on the size of the configuration.

Stability of the triangular Lagrange points beyond Gascheau’s value

“…Note that further studies on the stabiltiy of the equilateral configuration as a function of κ can be found in Santoprete (2006). The value of μ G = m 2 /(m 1 + m 2 ) = (1 − √ 23/27)/2 = 0.03852 .…”

Combined Effects of Radiation and Oblateness on the Existence and Stability of Equilibrium Points in the Perturbed Restricted Four-Body Problem

“…The equations of motion of the infinitesimal mass in the rotating coordinate system are written as (see, for example, Papadouris and Papadakis, [17] 2 ,…”

“…Later, Routh [19] studied the linear stability of the same solutions in the case of homogeneous potentials. Recently, Papadouris and Papadakis [17] produced the necessary condition for the stability of the Lagrange central configuration in the photogravitational R4BP based on the ideas of Santroprete [20] and Moeckel [21]. Based on these ideas, if someone replaces the masses…”

“…For example, Ceccaroni and Biggs [16] investigated the stability of the problem and tested the results in a real Sun-Jupiter-(624) Hektor-spacecraft system, Baltagiannis and Papadakis [17] studied periodic solutions of the problem in the SunJupiter-Trojan Asteroid-spacecraft system.…”

Planar N-body central configurations with a homogeneous potential

“…The general problem was made precise by Andoyer [12]. Particular cases have been studied for three bodies [50,117,115,121], four bodies [111,22,116,135], restricted cases (i.e. with one or more infinitesimal masses) [107], and polygonal and N + 1 ring systems [53,125,96,114,26,19]; more could be done, especially numerically, in our setting of a variable exponent potential.…”