Stability map of few-body configurations: Routh, Maxwell, hierarchy, and dissipation

We verify whether a simple symplectic integrator reproduces, without prior information, the classical stability criteria for few-body configurations. The Gascheau-Routh limit for the Lagrange triangle (μ ≈ 0.0198 for masses 1:μ:μ), the Maxwell criterion for rings with a central body (n ≥ 7 and μ < 2.298/n³), and the hierarchical criterion of Mardling and Aarseth are reproduced, the last with good agreement for m₃ ≥ 1. Within the linearly stable region, we document islands of instability associated with the 2:1 resonance between the eigenfrequencies, at β ≈ 0.0237 to 0.0239, in agreement with Deprit and Deprit-Bartholomé (1967). A linear drag destroys the gyroscopic stability of the triangle in a time approximately proportional to 1/γ, as predicted by the Kelvin-Tait-Chetaev theorem.