Hori method for generalized canonical systems

In this paper, some special features on the canonical version of Hori method, when it is applied to generalized canonical systems (systems of differential equations described by a Hamiltonian function linear in the momenta), are presented. Two different procedures, based on a new approach for the integration theory recently presented for the canonical version, are proposed for determining the new Hamiltonian and the generating function for systems whose differential equations for the coordinates describe a periodic system with one fast phase. These procedures are equivalent and they are directly related to the canonical transformations defined by the general solution of the integrable kernel of the Hamiltonian. They provide the same near-identity transformation for the coordinates obtained through the non-canonical version of Hori method. It is also shown that these procedures are connected to the classic averaging principle through a canonical transformation. As examples, asymptotic solutions of a non-linear oscillations problem and of the elliptic perturbed problem are discussed.