Notes on Hori Method for Canonical Systems

In this paper a slightly different approach is proposed for the process of determining the functions S_m and H_m^* of the algorithm of the canonical version of Hori method. This process will be referred to as integration theory of the mth order equation of the method. It will be shown that the ordinary differential equation with an auxiliary parameter t^* as independent variable, introduced through Hori auxiliary system, can be replaced by a partial differential equation in the time t. In this way, the mth order equation of the algorithm assumes a form very similar to the one of other perturbation methods. In virtue of this new approach of the integration theory for Hori method, Lagrange's variational equations introduced by Sessin are revised. As an example, the Duffing equation is solved through this new approach.