Stable and random motions in dynamical systems. With special emphasis on celestial mechanics. With a new foreword by Philip J. Holmes. 3rd pbk-ed. 3rd pbk-ed.

*(English)*Zbl 0991.70002
Princeton, NJ: Prineton University Press. 198 p. (2001).

As is well known by all the scientific community, J. Moser was one of the world’s leading mathematicians who helped develop theories in celestial mechanics, partial differential equations, functional analysis, complex geometry, calculus of variations, and is renowned for his work on the Kolmogorov-Arnold-Moser theory (KAM theory). Many dynamical systems have a very unstable (even ergodic) behavior, but there are also dynamical systems exhibiting stable behavior.

This book of Moser is mainly devoted to the study of this problem: how to decide when a system has stable or unstable behavior? With the help of integrable dynamical systems, it is possible to establish the stability of systems sufficiently close to these integrable systems. An example is, of course, the solar system where the mass ratios between the planets and the Sun are very small, so the forces between the planets are smaller than those between the planets and the Sun. Therefore, as a first approximation, we obtain the exactly integrable problem of non-interacting planets around the Sun where each planet will follow an elliptic orbit, and we will have quasi-periodic solutions considering the system as a whole. The main question arises whether one can find such quasi-periodic solutions for small but positive values of the masses.

This is precisely the celebrated problem of constructing quasi-periodic solutions (that is, invariant tori) in the \(N\)-body problem. Many methods has been devised for calculating these perturbations of the integrable case in celestial mechanics. The difficulty with these methods is that they lead to divergent series (as Weierstrass mentioned in 1878). The hardness of the convergence of these series is connected with the occurrence of “small divisors”, that is, rational dependence of frequencies of unperturbed motions by which it is necessary to divide in calculating the influence of perturbations. Obviously, this characteristic appears also in dynamical systems close to an integrable one. Then, more generally, one studies perturbations of quasi-periodic solutions in a system given by the Hamiltonian \(H=H_0+\varepsilon H_1\), where \(H_0\) is an integrable Hamiltonian. This problem was called the fundamental problem of dynamics by H. Poincaré. In such a case, these series expansions converge and represent bona fide solutions of the problem at least if certain Hessian determinant does not vanish.

The stability problem is the topic considered and beautifully analyzed by Moser in chapter II, where existence theorems for quasi-periodic solutions are derived. But also there are classes of solutions whose behavior is quite random, this question is related with the ergodic or transitive behavior of Hamiltonian systems of differential equations. In chapter III it is shown that quasi-periodic behavior on a set of positive measure and transitivity on other sets may coexist, even for analytical systems of differential equations. A carefully study of Sitnikov and Alekseev’s example is given. This example describes the motion in the restricted three-body problem, and a continuum of quasi-random solutions are exhibited. Therefore, chapter III is mainly devoted to the statistical behavior of solutions. Next, chapter IV contains some concluding remarks where, moreover, Moser proposes, as future work, the extension of these results to PDE; it is remarkable since Moser extended in 1988 KAM theory to nonlinear partial differential equations also to construct quasi-periodic solutions. Chapter V deals with technical details concerning small divisors, and chapter VI contains the technical proofs of the results of chapter III.

Concluding, this monograph, which contains the five Hermann Weyl lectures given by Moser at the Institute for Advanced Studies in Princeton [Stable and random motion in dynamical systems. With special emphasis on celestial mechanics. Hermann Weyl Lectures. The Institute for Advanced Study. Annals of Mathematics Studies. No. 77. Princeton, N. J.: Princeton University Press and University of Tokyo Press. VIII (1973; Zbl 0271.70009)], includes many of the most important steps in the development of stability problems for Hamiltonian systems, in particular, and in the qualitative theory of dynamical systems, in general.

This book of Moser is mainly devoted to the study of this problem: how to decide when a system has stable or unstable behavior? With the help of integrable dynamical systems, it is possible to establish the stability of systems sufficiently close to these integrable systems. An example is, of course, the solar system where the mass ratios between the planets and the Sun are very small, so the forces between the planets are smaller than those between the planets and the Sun. Therefore, as a first approximation, we obtain the exactly integrable problem of non-interacting planets around the Sun where each planet will follow an elliptic orbit, and we will have quasi-periodic solutions considering the system as a whole. The main question arises whether one can find such quasi-periodic solutions for small but positive values of the masses.

This is precisely the celebrated problem of constructing quasi-periodic solutions (that is, invariant tori) in the \(N\)-body problem. Many methods has been devised for calculating these perturbations of the integrable case in celestial mechanics. The difficulty with these methods is that they lead to divergent series (as Weierstrass mentioned in 1878). The hardness of the convergence of these series is connected with the occurrence of “small divisors”, that is, rational dependence of frequencies of unperturbed motions by which it is necessary to divide in calculating the influence of perturbations. Obviously, this characteristic appears also in dynamical systems close to an integrable one. Then, more generally, one studies perturbations of quasi-periodic solutions in a system given by the Hamiltonian \(H=H_0+\varepsilon H_1\), where \(H_0\) is an integrable Hamiltonian. This problem was called the fundamental problem of dynamics by H. Poincaré. In such a case, these series expansions converge and represent bona fide solutions of the problem at least if certain Hessian determinant does not vanish.

The stability problem is the topic considered and beautifully analyzed by Moser in chapter II, where existence theorems for quasi-periodic solutions are derived. But also there are classes of solutions whose behavior is quite random, this question is related with the ergodic or transitive behavior of Hamiltonian systems of differential equations. In chapter III it is shown that quasi-periodic behavior on a set of positive measure and transitivity on other sets may coexist, even for analytical systems of differential equations. A carefully study of Sitnikov and Alekseev’s example is given. This example describes the motion in the restricted three-body problem, and a continuum of quasi-random solutions are exhibited. Therefore, chapter III is mainly devoted to the statistical behavior of solutions. Next, chapter IV contains some concluding remarks where, moreover, Moser proposes, as future work, the extension of these results to PDE; it is remarkable since Moser extended in 1988 KAM theory to nonlinear partial differential equations also to construct quasi-periodic solutions. Chapter V deals with technical details concerning small divisors, and chapter VI contains the technical proofs of the results of chapter III.

Concluding, this monograph, which contains the five Hermann Weyl lectures given by Moser at the Institute for Advanced Studies in Princeton [Stable and random motion in dynamical systems. With special emphasis on celestial mechanics. Hermann Weyl Lectures. The Institute for Advanced Study. Annals of Mathematics Studies. No. 77. Princeton, N. J.: Princeton University Press and University of Tokyo Press. VIII (1973; Zbl 0271.70009)], includes many of the most important steps in the development of stability problems for Hamiltonian systems, in particular, and in the qualitative theory of dynamical systems, in general.

Reviewer: David Martin de Diego (Madrid)

##### MSC:

70-02 | Research exposition (monographs, survey articles) pertaining to mechanics of particles and systems |

70F15 | Celestial mechanics |

37N05 | Dynamical systems in classical and celestial mechanics |

37Cxx | Smooth dynamical systems: general theory |