Solvable Structures for Hamiltonian Systems

The paper proves a canonical solvable structure for completely integrable Hamiltonian systems by passing to action–angle variables and constructing G_i via a linear-ansatz Gi = Σ_j g_{ij}(P) Q_j; the required commutators are enforced by solving linear PDEs and linear systems using the invertibility of DF, culminating in Theorem 3.1 (see (14)–(16), (17)–(21), (27)–(31), and Theorem 3.1). The candidate solution instead applies the Carathéodory–Jacobi–Lie theorem to choose local canonical coordinates (F,G) with {F_i,G_j}=δ_ij and {G_i,G_j}=0, which yields a valid special case of the paper’s structure with f^l_{ij}=0 and h_{il}=−∂^2H/∂F_i∂F_l. These two approaches are compatible: the paper allows nonzero f^l_{ij} in general, while the model realizes a particular choice where they vanish. Theorem 3.1 and its construction in the paper support the model’s argument as a legitimate local instance of the canonical solvable structure .