Introduction

Abstract

In what follows we are going to describe in short the basic problem of perturbations the way it is to be developed in these notes. We shall here make free and simple statements without entering mathematical details on the functions involved. The necessary hypotheses will be made in the subsequent chapters. Historically we consider Lindstedt’s (l882) problem of obtaining a series solution, free from secular and/or mixed secular terms, of the equation

$$ \ddot{x} + \omega_0^2x = \in f(x,\dot{x},t) $$

where 0 < ∈ < 1 is a parameter. The possibility of obtaining a solution

$$ x = {x_0}(t) + \in {x_1}(t) + { \in^2}{x_2}(t) + ... $$

$$ \dot{x} = {\dot{x}_0}(t) + \in {\dot{x}_1}(t) + { \in^2}{\dot{x}_2}(t) + ... $$

of the above equation, with x_j(t), ẋ_j(t) bounded functions for all t ∈ R was found to depend essentially on the nature of f and its derivatives up to some order. The reference solution introduced by Lindstedt, that is, (x_o (t), ẋ_o (t)) was given by

$$ {x_0} = a\,\cos (\omega t + \sigma ) $$

$$ {\dot{x}_0} = - a\omega \,\sin (\omega t + \sigma ) $$

where ω is a priori unknown but, by assumption, developable in a power series

$$ \omega = {\omega_0} + \in {\omega_1} + { \in^2}{\omega_2} + { \in^3}{\omega_3} + ... $$

where ω₁, ω₂,… are constants depending on ω_o, a and f.