Stable motions of high energy particles interacting via a repelling potential

The paper proves that, under SP1–SP3 for the single-particle orbit and IP1 for the averaged interaction, there exists a positive-measure set of quasi-periodic choreographic motions near L* with uniform O(δ^{1/4}) closeness and frequency shift O(δ^{1/2}); this is established via a careful scaling/averaging scheme, comparison of the Poincaré map with the time-1 map of an explicit near-integrable Hamiltonian, and a KAM argument that yields maximal tori of positive measure on each small energy level (Theorem 1 and Section 3) . The candidate solution reproduces much of the high-level strategy (normal form, mean/relative phases, averaging, scaling, KAM), but it only constructs lower-dimensional tori by setting the normal and mean-difference blocks to zero and appealing to Pöschel’s lower-dimensional KAM, then explicitly states positive measure only relative to a tangential slice (and only in full phase space when d=1). This falls short of the paper’s stronger claim of positive measure on energy levels in general d (cf. the proof summary around system (54)→(55)→(58) and Lemma 3.4) . In addition, the model’s invocation of lower-dimensional KAM elides required global nonresonance conditions with the O(1) normal spectrum beyond the low-order SP2 hypothesis; the paper circumvents this by constructing a near-identity normal form whose time-1 map satisfies a standard twist nondegeneracy (det A_ω ≠ 0) sufficient for maximal KAM tori (Lemma 3.4) . Hence the paper’s result is correct and stronger, while the model’s proof is incomplete for the stated measure claim.