Global Weinstein Type Theorem on Multiple Rotating Periodic Solutions for Hamiltonian Systems

The paper proves Theorem 1.1 (at least n geometrically distinct Q-rotating periodic solutions on a Q-invariant, strictly convex energy surface under the pinching R < sqrt(2) r) by introducing a new Q(s)-index, reducing to a q-homogeneous Hamiltonian with 1<q<2 on the same energy surface, and applying a Legendre-transform-based functional E that is Q(s)-invariant and satisfies Palais–Smale; the abstract index theory then yields n distinct Q(s)-critical orbits corresponding to rotating solutions . By contrast, the model’s solution hinges on choosing a 2-homogeneous Hamiltonian and a Clarke–Ekeland functional on the twisted loop space, then invoking a Maslov-type index “window.” This conflicts with the paper’s critical step that requires p = q/(q−1) > 2 (hence 1<q<2) to ensure coercivity, PS, and negativity estimates that position minimax levels below m*, securing minimal Q-rotating period T and multiplicity; the model’s q=2 choice forces p=2, undermining these core estimates and the minimax scheme used in the paper . The model also leans on a spectral/Maslov index “window” not established in the paper’s framework and leaves key steps (PS on the twisted space with q=2, geometric distinctness under Q-rotation) unproved.