New method of investigating asymptotic expansions

Methods of integrating systems written in Hamiltonian form are considered. On the basis of the method of canonical transformations, a general solution is obtained. It is shown that an arbitrary function of the coordinates and the momenta, depending on the time in view of the system, may bewritten in the form of P-order edexponentials. As an example, the solution of equations for Airey functions is considered. An algorithm of canonical transformation to mean variables and the Hamiltonian of the mean motion is considered. In the last section, a new derivation is obtained for the equations determining the eigenfunctions and eigenvalues. An algorithm is presented for the solution of the eigenvalue problem and the calculation of an arbitrary function of the matrix elements in all orders with respect to the small parameter.