Nonlinear Coupled Oscillators. I. Perturbation Theory; Ergodic Problem

A study is made of some of the problems which arise in determining the long‐time behavior of a system of coupled oscillators. Standard perturbation methods are examined in the light of certain classic results due to Poincaré and Whittaker concerning the construction of constants of motion which are analytic in the coupling constant, λ. These considerations lead to the study of perturbation methods which are not ordered in powers of λ. An examination of the various advantages of these methods leads to a method which removes secular terms in such a way as to mitigate the classic problem of small divisors. The significance of studies which attempt to relate the existence of analytic constants of the motion with the ergodic behavior of a system is examined.