NONLINEAR STABILITY OF RELATIVE EQUILIBRIA IN PLANAR N-VORTEX PROBLEM

The paper’s main theorem (Theorem 5.2) establishes an Energy–Casimir-type Lyapunov stability criterion for fixed points of the Lie–Poisson relative dynamics constrained to the rank-one invariant manifold R^{-1}(0), using f = a0 h + Σ ai Ci plus constraint terms, with Df(µ0)=0 and a positive-definite Hessian on T_{µ0}M; the proof relies on (i) invariance of v̊(1)_K = R^{-1}(0) and the Casimirs Cj(µ)=tr((iKµ)^j) for (14a) and (ii) a stability lemma adapted from Aeyels together with second-order constrained optimization to show local uniqueness of the invariants’ level set in R^{-1}(0) . The candidate solution implements the same Energy–Casimir-with-constraints method: it constructs f with R-terms, verifies invariance of the constraint set and of the chosen invariants, imposes Df(µ0)=0 and H>0 on T_{µ0}M, and then argues Lyapunov stability via level sets of f. The only substantive flaw is Step 1, which asserts a standard Lax equation for L=iKµ; the evolution (14a) does not yield a simple [B,L]-commutator for that L, and calling the resulting tr((iKµ)^j) “Casimirs” based solely on that Lax claim is not justified. The paper instead proves Cj are Casimirs for the Lie–Poisson bracket and proves R−1(0) invariance by the structure µ=izz*, or equivalently by the flow staying in v̊(1)_K . Once this minor correction is made (e.g., using µ̇=[(δh/δµ)K^{-1},µ] to preserve rank and using the Casimir property from the Poisson structure to preserve Cj), the model’s argument aligns with the paper’s theorem and proof strategy.