Introduction

Abstract

In this text on nonlinear physics, we are primarily interested in the problem of how to deal with physical phenomena described by nonlinear ordinary or partial differential equations (ODEs or PDEs), i.e., by equations which are nonlinear functions of the dependent variables. For the familiar simple pendulum (Figure 1.1) of classical mechanics, a mass m attached to a rigid massless rod with a length ℓ, the relevant equation of motion is

$$ \ddot{\theta } + \omega _{0}^{0}\sin \theta = 0 $$

(1.1)

with ${{\omega }_{0}} = \sqrt {{g/\ell }}$, g being the acceleration due to gravity, and dots denoting derivatives with respect to time. The term sin θ is a nonlinear function of θ. In elementary physics courses, one limits the angle θ to sufficiently small values, so that sin θ≃θ, and Equation (1.1) reduces to the linear simple harmonic oscillator equation,

$$ \ddot{\theta } + \omega _{0}^{2}\theta = 0. $$

(1.2)

In the beginning God created the heavens and the earth, the earth was at first a shapeless, chaotic mass,...Genesis 1: The Way (The Living Bible), 1972