Nearly-periodic maps and geometric integration of noncanonical Hamiltonian systems

The paper proves, for non-resonant presymplectic nearly-periodic maps on exact presymplectic manifolds, the existence of a U(1)-invariant primitive θ_γ and an adiabatic invariant μ_γ = ι_{R_γ}θ_γ, with two sufficient hypotheses: (1) the map is Hamiltonian, or (2) M is connected and R_0 has a zero. The construction of θ_γ by averaging over the roto-rate and the Hamiltonian case proof for μ_γ are correct and align with the candidate’s approach. However, for the non-Hamiltonian case (condition (2)), the model’s proof is incomplete: it does not invoke the paper’s essential normalizing transformation argument to force c_γ = F^*_γ μ_γ − μ_γ ≡ 0, and its order-by-order argument at a zero of R_0 omits necessary justification. The paper’s proof under (2) is complete and correct, while the model’s is not. Key steps are supported by Theorem 3 (roto-rate is presymplectic), the averaging construction of θ_γ, and Theorem 4’s handling of c_γ in both cases, especially the normalizing transformation in case (2) .