Frequency Coalescence as a Conservative Early Warning: Krein Collisions at the Triangular Lagrange Points and in the Penning Trap

This paper identifies a conservative analogue of critical slowing down. In dissipative systems the approach to a saddle-node is signalled by a vanishing recovery rate. Hamiltonian systems lack any such rate: an elliptic equilibrium has purely imaginary eigenvalues, so the usual diagnostics are identically zero. The author therefore studies the approach to a Hamiltonian–Hopf bifurcation—the collision of two elliptic modes of opposite Krein signature—in two-degree-of-freedom linear Hamiltonian systems. The squared frequency gap supplies an exact, coordinate-invariant diagnostic whose vanishing marks the collision. From this identity a compact set of scaling laws follows: the gap closes with the square root of the distance to the critical parameter, the growth rate beyond the collision mirrors half the pre-collision gap, the eigenvector condition number and maximal transient amplification both diverge, and the mean-square displacement under additive white noise grows at a rate that likewise diverges. Full-depth beats appear in generic observables, and records shorter than a diverging multiple of the beat period cannot distinguish a stable system from one already at the collision. These statements are applied to the triangular Lagrange points at the Routh mass ratio and to the radial motion of an ideal Penning trap—both gyroscopically stabilised potential maxima. Explicit constants are computed, and numerical integration of the planar restricted three-body problem confirms the linear predictions while also revealing that the gap widens with amplitude. Just beyond the critical ratio, orbits remain bounded on the integration times examined. The results are carefully localised: they apply only to opposite-signature collisions and give no warning of low-order resonance instabilities that occur at other mass ratios. The paper’s principal strengths are its precision and its restraint. The scaling laws are derived jointly rather than piecemeal, the constants for the two physical examples are given explicitly, and the observational cost of the warning is quantified. Limitations—locality, coordinate dependence of the prefactors, the absence of damping, and the specificity to one exit route—are stated clearly. The placement within the author’s earlier “Trojan Universality Class” taxonomy is consistent, though the paper itself remains a self-contained contribution to early-warning diagnostics for conservative systems. The work is technically careful and conceptually economical.