CONSTRUCTIVE QP-TIME-DEPENDENT KAM ALGORITHM FOR LAGRANGIAN TORI

The paper derives the triangularized linearized cohomological system in an adapted symplectic frame P = (L N), namely Λ ξ + L_{ω,α} ξ = Ω₀ P^⊤ Ω(K) E with Λ = [[0, T]; [0, 0]], which splits into L_{ω,α} ξ_N = η_N and L_{ω,α} ξ_L + T ξ_N = η_L, and then solves via the Fourier right-inverse R_{ω,α} under the conditions 〈η_N〉 = 0 and det〈T〉 ≠ 0; see equations (2.23)–(2.26) and Algorithm 2.4 in the PDF . The candidate solution reproduces the same derivation and formulas in the canonical symplectic case (Ω = Ω₀), including the definition of the torsion via Th = Ω₀ DzZh − DzZh Ω₀ and T = N^⊤ Th N, and uses the same right-inverse R_{ω,α} for mean-zero functions (paper’s (2.19)) . Hence both are consistent and effectively the same proof specialized to the canonical case.