How Weak Dissipation Destabilizes Gyroscopic Equilibria: Krein Signature, the Redistribution of Damping, and Amplification near Frequency Collision

Doucette’s second paper quantifies a classical fragility: an equilibrium sitting at a potential maximum can be linearly stable under gyroscopic forces alone, yet the smallest dissipation destroys that stability. The Kelvin–Tait–Chetaev theorem is qualitative; this work supplies rates for linear two-degree-of-freedom systems. Closed-form expressions show that the slow mode always carries negative energy when the potential is a maximum, and that every nonzero positive-semidefinite damping drives it unstable. The growth rate is bounded from below by an explicit positive constant times the damping coefficient, so no direction of damping is harmless—only relatively less harmful. An exact sum rule further shows that damping is redistributed rather than removed: the total contraction of the two modes is fixed, so rapid decay of the fast mode can mask growth of the slow one.Near a frequency collision the first-order rate diverges. A local crossover formula captures the transition from linear growth in the damping coefficient to a square-root saturation once the damping becomes comparable to the square of the frequency gap; both the triangular points and the ideal Penning trap collapse onto the same scaling curve. Explicit results follow for the triangular Lagrange points under drag relative to the rotating frame (radial damping is nearly innocuous, tangential damping is not), for a medium at rest (the system becomes conformally symplectic and every eigenvalue acquires the same real part), and for a rigidly rotating medium (a sharp threshold rotation rate separates stabilization from destabilization). In the ideal Penning trap the magnetron mode grows for every isotropic damping, with a closed-form spectrum.Nonlinear integrations of the planar restricted three-body problem recover the linear rates to high accuracy at small amplitude. The paper is precise about its scope: the formulas are first-order in the damping, restricted to two degrees of freedom, and the nonlinear checks are limited. Within the author’s Trojan Universality Class the results identify the negative-energy mode as the common ingredient of three distinct exit routes. The contribution is quantitative rather than qualitative, and the limitations are stated with the same care as the theorems.