McGehee blowup for Lagrangian systems and instability of equilibria

The paper proves a generic total-instability theorem for real-analytic electromagnetic Lagrangians under the weak magnetism hypothesis d − l/2 ≥ 1 (with l the first nonzero jet degree of U and d that of µ), culminating in Theorem 1.1 / 11.4: there exists an open and dense subset A ⊂ Ik of potential germs for which p is totally unstable . The candidate claims a stronger statement—taking A = Ik—and argues via a virial function and a purported uniform “radial Lojasiewicz” inequality. The key defect is the control of the magnetic term ⟨x, Y(v)⟩: the candidate bounds it by ε(−U) uniformly on the subcritical set by using only −U ≤ C|x|^l. This direction of inequality is insufficient. When U<0 first occurs at order m>l along some directions (i.e., U_l vanishes on those directions), −U can scale like |x|^m, so |⟨x,Y(v)⟩| ∼ |x|^{d} |v| ≤ C |x|^{d} (−U)^{1/2} and the ratio |⟨x,Y(v)⟩|/(−U) behaves like |x|^{d − m/2}, which can blow up as |x|→0 if m/2>d. Thus the claimed bound |⟨x,Y(v)⟩| ≤ ε(−U) fails in general; the virial second-derivative F'' need not be positive. The paper avoids exactly this pitfall via the McGehee blowup and generic structure on the boundary ∂M≤0, which requires f (the restriction of Ul to the unit sphere) to be Morse with 0 a regular value, ensuring uniform control of the boundary dynamics and yielding total instability generically (Proposition 11.1, Corollary 11.3, Theorem 11.4) . The model’s claim that A=Ik contradicts the paper’s carefully stated genericity and omits these necessary hypotheses.