Introduction to Hamiltonian Dynamical Systems and the N-Body Problem

In this chapter we study Hamiltonian systems which are linear differential equations. Many of the basic facts about Hamiltonian systems and symplectic geometry are easy to understand in this simple context. The basic linear algebra introduced in this chapter is the cornerstone of many of the later results on nonlinear systems. Some of the more advanced results which require a knowledge of multilinear algebra or the theory of analytic functions of a matrix are relegated to the appendices or references to the literature. These results are not important for the main development.

Differential forms play an important part in the theory of Hamiltonian systems, but his theory is not universally known by scientists and mathematicians. It gives the natural higher-dimensional generalization of the results of classical vector calculus. We give a brief introduction with some, but not all, proofs and refer the reader to Flanders (1963) for another informal introduction but a more complete discussion with many applicatons, or to Spivak (1965) or Abraham and Marsden (1978) for more complete mathematical discussion. The reader conversant with the theory of differential forms can skip this chapter, and the reader not conversant with the theory should realize that what is presented here is not meant to be a complete development but simply an introduction to a few results that will be used sparingly later.

The form of Hamilton’s equations is very special, and the special form is not preserved by an arbitrary change of variables; so, the change of variables that preserve that special form are very important in the theory. The classical subject of celestial mechanics is replete with special coordinate systems which bear the names of some of the greatest mathematicians. We shall consider some of them in this chapter.

This chapter gives an introduction to the geometric theory of autonomous Hamiltonian systems by studying some local questions about the nature of the solutions in a neighborhood of a point or a periodic solution. The dependences of periodic solutions on parameters is also presented in the case when no drastic changes occur, i.e., when there are no bifurcations. Bifurcations are addressed in Chapter VIII. Several applications to the 3-body problem are given. The chapter ends with a brief introduction to hyperbolic objects and homoclinic phenomena.

In the last chapter, some local results about periodic solutions of Hamiltonian systems were presented. The systems contain a parameter, and the conditions under which a periodic solution can be continued in the parameter were discussed. Since Poincaré used these ideas extensively, it has become known as Poincaré’s continuation method. By Lemma V.E.2, a solution ø(t, ξ′) of an autonomous differential equation is T-periodic if and only if ø(T’) = ø is the general solution. This is a finite-dimensional problem since is a function defined in a domain of ℝ^m+1 into ℝ^m. Thus, periodic solutions can be found by the finite-dimensional methods, i.e., the finite-dimensional implicit function theorem, the finite-dimensional fiixed point theorems, the finite-dimensional degree theory, etc. This chapter will present results which depend only on the finite-dimensional implicit function theorem. Chapter X will present a treatment of fixed point methods as they apply to Hamiltonian systems. In this chapter the periodic solutions vary continuously with the parameter (“can be continued”), but Chapter VII will discuss the bifurcations of periodic solutions.

Perturbation theory is one of the few ways that one can bridge the gap between the behavior of a real nonlinear system and its linear approximation. Because the theory of linear systems is so much simpler, investigators are tempted to fit the problem at hand to a linear model without proper justification. Such a linear model may lead to quantitative as well as qualitative errors. On the other hand, so little is known about the general behavior of a nonlinear system that some sort of approximation has to be made.

This chapter and Chapter IX use the theory of normal forms developed in Chapter VI. They contain an introduction to generic bifurcation theory and its applications. Bifurcation theory has grown into a vast subject with a large literature; so, this chapter can only present the basics of the theory. The primary focus of this chapter is the study of periodic solutions—their existence and evolution. Periodic solutions abound in Hamiltonian systems. In fact, a famous Poincaré conjecture is that periodic solutions are dense in a generic Hamiltonian system, a point that was established in the C ¹ case by Pugh and Robinson (1977).

Questions of stability of orbits have been of interest since Newton first set down the laws that govern the motion of the celestial bodies. “Is the universe stable?” is almost a theological question. Even though the question is old and important, very little is known about the problem, and much of what is known is difficult to come by.

In this chapter we study the dynamics of area-preserving (i.e., symplectic) monotone twist maps of the annulus. While seemingly quite special, we have already seen examples of such maps as time one maps of time periodic Hamiltonian systems of one degree of freedom, as Poincaré section maps for periodic orbits of Hamiltonian systems of two degrees of freedom (see Chapter V, Sections B and E). These maps also appear as dynamical systems in their own right (e.g., biliards on a convex table and one-dimensional crystals; see V.B).