Chapter 68 - George David Birkhoff, Dynamical systems (1927)

Chaos is unpredictable behaviour arising in systems governed by deterministic laws. Since its ‘discovery’ in the 1960s by Edward Lorenz in his studies of meteorological systems, chaos has become an important way of understanding seemingly random behaviour in many other areas. These include the double pendulum, the climate, El Niño and electrical oscillators. Understanding chaos allows the limits of the predictability of a system to be determined. However, despite this recent activity this paper will show that the theory of chaos has a rich history and chaotic behaviour was originally identified in the 1890s by Henri Poincaré in his investigations of the three-body problem in celestial mechanics. But remarkably this paper will also show that chaotic behaviour can be found in such a simple problem as multiplication by 10 and it is closely linked to the question (studied by the ancient Greeks) of whether a number is rational or irrational.

Analytical solutions to a variety of simplified versions of the restricted three-body problem in celestial mechanics possess long running history that encompasses several centuries. Most of the successes were limited either to the planar configuration of the three bodies, to the motion around the Lagrange points, or to the Kozai–Lidov effect. We review some analytical advances obtained by separating rapid and slow subsystems as presented in recently published papers concerning the non-planar motion of the three bodies unrelated to the Lagrange points and to the Kozai–Lidov effect. Most (but not all) of the discussed advances correspond to the bound motion in the considered celestial systems.

It was a surprise to me that I would have to present a paper at the final session of the Congress in this large hall, which had been known to me rather as a place for the performance of great musical compositions of the world conducted by Mengelberg. The paper which I have prepared, without taking into account that it would occupy such an honourable position in the programme of the Congress, is devoted to a rather special range of problems. My aim is to elucidate ways of applying basic concepts and results in the modern general metrical and spectral theory of dynamical systems to the study of conservative dynamical systems in classical mechanics. However, it seems to me that the subject I have chosen may also be of broader interest, as one of the examples of the appearance of new, unexpected and profound relationships between different branches of classical and modern mathematics.

A class of periodic solutions for the restricted three-body problem

Explore the restricted three-body problem in three dimensions—a classic problem in celestial mechanics—by varying the initial positions and velocities of a particle moving around two fixed masses, whose masses are and .

Several examples of invariant manifold problems in celestial mechanics are discussed including the determination of the topological character of the Kepler energy manifold and the derivation of a priori bounds for the artificial satellite problem.

In this chapter we introduce one-dimensional dynamical systems and analyze some elementary examples. A study of the iteration in Newton's method leads naturally to the notion of attracting fixed points for dynamical systems. Newton's method is emphasized throughout as an important motivation for the study of dynamical systems. The chapter concludes with various criteria for establishing the stability of the fixed points of a dynamical system.

In the following paper we tried to apply the Lie-formalism to the regularized restricted three body problem. It will be shown that this algorithm leads to a very simple structure program which is also fast.

On certain determinants connected with a problem in celestial mechanics

We present an algorithm to compute the incomplete elliptic integral of a general form. The algorithm efficiently evaluates some linear combinations of incomplete elliptic integrals of all kinds to a high precision. Some numerical examples are given as illustrations. This enables us to numerically calculate the values and the partial derivatives of incomplete elliptic integrals of all kinds, which are essential when dealing with many problems in celestial mechanics, including the analytic solution of the torque-free rotational motion of a rigid body around its barycenter.

The evolution of the solar system is an interesting dynamical problem in celestial mechanics. Computer simulations have shown that planetary bodies in multiple-body systems become unstable after a long time. In this thesis, through numerical simulations, we investigate the stability of the Earth in a binary star system. In the Sun, Earth, and Jupiter three-body system, we treat Jupiter as the second star (Star 2) and we simulate the Earth's orbit over many orbital periods. In our program, we generate the positions, velocities, accelerations, and other orbital elements of each body using Newton's laws of motion. We explore four types of simulations including when the system is run with the usual locations and masses, when the mass of Star 2 is changed at the position of Jupiter, when the stars have equal mass and the location of Star 2 is changed, and when the position and mass of Star 2 changed. We plot the position coordinates of the Earth and determine when the Earth exhibits unpredictable or erratic behaviors.

Application of Lie-Series to Regularized Problems in Celestial Mechanics

The purpose of this series of lecture notes is to give an outline of the basic tools required to show the occurrence of chaotic motions in the simplest non--integrable problems in Celestial Mechanics, such as the circular restricted planar 3--body problem. No formal proofs will be given here; they can be found in the references given for each section. Section 1 describes the linear and local theory of ordinary differential equations in the neighbourhood of a fixed point; the problems arising in the embedding of invariant stable and unstable manifolds are also discussed. Section 2 is about periodic orbits; the subjects discussed include variational equations, surfaces of section, the continuation of periodic orbits in the restricted 3--body problem, and bifurcation of hyperbolic periodic orbits from resonant periodic orbits. Section 3 covers fundamental models of resonance, the global behaviour of separatrices, and their intersections; all this allows to give at least an outline of the proof of the fundamental result --presented by Poincaré in his book Les méthodes nouvelles de la mécanique céleste-- by which homoclinic points must necessarily occur in the restricted problem. A short conclusion underscores the fact --shown in a rigorous vay much later-- that this in turn implies that chaos in the strongest possible sense occurs in the restricted problem, and is an essential feature of every non--integrable system, even very simple ones with only two degrees of freedom. I apologize for reporting here my lectures in a very short format, almost without comments in between the formulas and statements of the main results; my understanding of the purpose of this notes is that they should serve as a reminder of the existence of many subjects to be studied, rather than a complete presentation which could not be contained in this format.

The determination of the motion of N point-like masses subject to their mutual gravitational forces is the fundamental problem in celestial mechanics, with important astrophysical applications, including the dynamics of planetary and satellite systems and the evolution of stellar systems (ranging from multiple stars to stellar clusters and galaxies). From the perspective of theoretical physics, this has always been a core problem in gravitation theory; through observations of the motion of celestial bodies it can be tested, quantitatively validated, or even disproved. However, even for N as small as 3, it is impossible to find general solutions in terms of simple analytical functions (as for the two-body problem). A great variety of orbits are possible and it can be shown that only particular initial conditions give rise to periodic behaviour. Even the perturbative techniques worked out in Ch. 12 can be successfully applied only in the particular case of hierarchical systems, where the attraction of one body prevails (see Sec. 15.1). As a consequence, the general gravitational N-body problem has been studied by two alternative methods which can yield only partial results, but are in some way complementary. The first is numerical integration, starting from given initial conditions. In this way the motion can be determined in detail, but only for a limited span of time (due to both limitations in computer time and accumulation of numerical errors; see Sec. 15.1); in addition, it is often impossible to determine the region of phase space where a property of the numerically determined orbits holds. The second method is the search for general constraints or criteria regarding qualitative features of the motion, such as its periodic character, its stability and its geometrical and topological properties. KeywordsEquilibrium PointSolar SystemOrbital PlaneTest MassTidal ForceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The dynamics of the spin–spin problem in Celestial Mechanics