When Koopman Meets Hamilton and Jacobi

The candidate reproduces the paper’s Procedure One step-by-step: (i) it solves the nominal PDE with W(x,t)=P^T e^{-Λ t} Φ(x) and uses W as a generating function to obtain p=(∂Φ/∂x)^T e^{−Λ^T t}P and X=e^{−Λ t}Φ(x), leading to complete integrability of H0 (matching Proposition 3 and Eq. (45) with Ẋ=Ṗ=0) ; (ii) under Approximation 1, it derives the same time-varying quadratic H̄1, performs the same symplectic rescaling (X̄,P̄)=(e^{Λ t}X,e^{−Λ^T t}P), and obtains the same LTI Hamiltonian system with matrix ℋ=[Λ −R1; −Q1 −Λ^T] (Eqs. (50)–(55)) ; (iii) it characterizes the stable Lagrangian subspace as the graph P̄=L X̄ with L=−D2^{-1}D1 and shows L solves the Riccati equation Λ^T L+LΛ−L R1 L+Q1=0 (Propositions 4–5), then pulls back to p=(∂Φ/∂x)^T L Φ(x) and V(x)=½ Φ^T L Φ(x) (Eqs. (56)–(60)) . The only minor issue is a sign slip in one line of the appendix (solving D1+D2L=0), but the stated proposition uses the correct sign; the candidate’s sign is correct as well . Overall, both arguments agree in content and structure.