NON-RESONANT CIRCLES FOR STRONG MAGNETIC FIELDS ON SURFACES

The paper establishes (via a rigorous Hamiltonian normal form and a Moser–twist/Poincaré–Birkhoff argument) that for every non-resonant magnetic system at zero speed there is a circle L and small speeds s for which (i) there are infinitely many periodic magnetic geodesics in any neighborhood of L and (ii) trajectories starting sufficiently close to L remain trapped; see Theorem 1.4 and its proof, together with the explicit normal form (4.1)–(4.2) and the twist-map mechanism in Section 3 . The candidate solution follows a similar high-level plan but makes two decisive mistakes. First, it identifies the twist/non-resonance condition with having nonzero normal derivative ∂_n H0 along L, which is not the correct criterion; in the paper the twist condition is f′(r0) ≠ 0 in action–angle coordinates (equation (3.3) and its use in Theorem 3.1), and non-resonance is guaranteed by constructing a non-resonant circle via Section 2, not by merely picking a regular level . Second, in the constant-b case the candidate asserts an O(s^2) effective drift, whereas the paper’s normal form shows the first nontrivial term appears at order s^4 and is proportional to K (equation (4.2)) . The model also over-claims that the boundary circles are rotated in opposite directions (unneeded and generally false; the paper requires only distinct boundary rotation numbers). Consequently, while the conclusions mirror the paper’s, the model’s proof hinges on incorrect criteria and scalings. The paper’s argument is correct; the model’s is not.