Integrable Systems

In this chapter we consider a special class of autonomous systems, $$x^\prime = X(x),$$ , on open sets $$\mathcal{O} \subseteq \mathbb{R}^n$$ , whose integral curves are completely “determined” by n - 1 functions, F 1, F 2, … , F n-1 : $$U \subseteq \mathcal{O} \to \mathbb{R}$$ , defined on an open dense subset U of O. These functions are called first integrals, or constants of the motion, and have, by definition, constant values along each integral curve of X. In addition, there are conditions on F 1, F 2, … , F n-1, so that the level sets F i(x) = k i, i = 1, … , n - 1, intersect to give 1-dimensional submanifolds or curves in ℝn and these curves coincide, in a sense, with the integral curves of X. Such systems are called integrable systems and will be defined more precisely below. Integrable systems are often called completely integrable systems in accordance with the terminology used in the more general subject of Pffafian systems (see [BCG 91], [Sl 70], [Di 74]). However, in the study of Hamiltonian systems (Chapter 9), there is the well-accepted term of completely integrable Hamiltonian system, which is related to but quite distinct from the type of system studied here. Thus, we will use the terms “integrable” and “completely integrable” to distinguish between the two distinct types of the systems studied in this chapter and in Chapter 9, respectively. This naming convention was suggested by Olver [Olv 96, p. 70].