Is the Earth’s orbit stable? Will the Moon always point the same face to our planet? Will some asteroid collide with the Earth? These questions have puzzled mankind since antiquity, and answers have been looked for over the centuries, even if these events might occur on time scales much longer than our lifetime. It is indeed extremely difficult to settle these questions, and despite all efforts, scientists have been unable to give definite answers. But the advent of computers and the development of outstanding mathematical theories now enable us to obtain some results on the stability of the solar system, at least for simple model problems.

The stability of the solar system is a very difficult mathematical problem, which has been investigated in the past by celebrated mathematicians, including Lagrange, Laplace and Poincaré. Their investigations lead to the development of *perturbation theories*—theories to find approximate solutions of the equations of motion. However, such theories have an intrinsic difficulty related to the appearance of the so-called *small divisors*—quantities that can prevent the convergence of the series defining the solution.

A breakthrough occurred in the middle of the 20th century. At the 1954 International Congress of Mathematics in Amsterdam, the Russian mathematician Andrei N. Kolmogorov (1903-1987) gave the closing lecture, entitled *“The general theory of dynamical systems and classical mechanics.”* The lecture concerned the stability of specific motions (for the experts: the persistence of quasi-periodic motions under small perturbations of an integrable system). A few years later, Vladimir I. Arnold (1937-2010), using a different approach, generalized Kolmogorov’s results to (Hamiltonian) systems presenting some degeneracies, and in 1962 Jürgen Moser (1928-1999) covered the case of finitely differentiable systems. The overall result is known as *KAM theory* from the initials of the three authors [K], [A], [M]. KAM theory can be developed under quite general assumptions.

An application to the N-body problem in Celestial Mechanics was given by Arnold, who proved the existence of some stable solutions when the orbits are nearly circular and coplanar. Quantitative estimates for a three-body model (e.g., the Sun, Jupiter and an asteroid) were given in 1966 by the French mathematician and astronomer M. Hénon (1931-2013), based on the original versions of KAM theory [H]. However, his results were a long way from reality; in the best case they proved the stability of some orbits when the primary mass-ratio is of the order of $10^{-48}$—a value that is inconsistent with the astronomical Jupiter-Sun mass-ratio, which is of the order of $10^{-3}$. For this reason Hénon concluded in one of his papers, *“Ainsi, ces théorèmes, bien que d’un très grand intérêt théorique, ne semblent pas pouvoir en leur état actuel être appliqués á des problèmes pratiques”* [H]. This result led to the general belief that, although an extremely powerful mathematical method, KAM theory does not have concrete applications, since the perturbing body must be unrealistically small. During one of my stays at the Observatory of Nice in France, I had the privilege to meet Michel Hénon. In the course of one of our discussions he showed me his computations on KAM theory, which were done by hand on only two pages. It was indeed a success that such a complicated theory could be applied using just two pages! Likewise, it was evident that to get better results it is necessary to perform much longer computations, as often happens in classical perturbation theory.

A new challenge came when mathematicians started to develop *computer-assisted proofs*. With this technique, which has been widely used in several fields of mathematics, one proves mathematical theorems with the aid of a computer. Indeed, it is possible to keep track of rounding and propagation errors through a technique called interval arithmetic. The synergy between theory and computers turns out to be really effective: the machine enables us to perform a huge number of computations, and the errors are controlled through interval arithmetic. Thus, the validity of the mathematical proof is maintained. The idea was then to combine KAM theory and interval arithmetic. As we will see shortly, the new strategy yields results for simple model problems that agree with the physical measurements. Thus, computer-assisted proofs combine the rigour of the mathematical computations with the concreteness of astronomical observations.

Here are three applications of KAM theory in Celestial Mechanics which yield realistic estimates. The extension to more significant models is often limited by the computer capabilities.

- A
*three-body problem*for the Sun, Jupiter and the asteroid Victoria was investigated in [CC]. Careful analytical estimates were combined with a Fortran code implementing long computations using interval arithmetic. The results show that in such a model the motion of the asteroid Victoria is stable for the realistic Jupiter-Sun mass-ratio. - In the framework of
*planetary problems*, the Sun-Jupiter-Saturn system was studied in [LG]. A bound was obtained on the secular motion of the planets for the observed values of the parameters. (The proof is based on the algebraic manipulation of series, analytic estimates and interval arithmetic.) - A third application concerns the rotational motion of the Moon in the so-called
*spin-orbit resonance*, which is responsible for the well-known fact that the Moon always points the same face to the Earth. Here, a computer-assisted KAM proof yielded the stability of the Moon in the actual state for the true values of the parameters [C].

Although it is clear that these models provide an (often crude) approximation of reality, they were analyzed through a *rigorous* method to establish the stability of objects in the solar system. The incredible effort by Kolmogorov, Arnold and Moser is starting to yield new results for concrete applications. Faster computational tools, combined with refined KAM estimates, will probably enable us to obtain good results also for more realistic models. Proving a theorem for the stability of the Earth or the motion of the Moon will definitely let us sleep more soundly!

References:

[A] V.I. Arnold, “Proof of a Theorem by A.N. Kolmogorov on the invariance of quasi-periodic motions under small perturbations of the Hamiltonian,” Russ. Math. Surveys, vol. 18, 13-40 (1963).

[C] A. Celletti, “Analysis of Resonances in the Spin-Orbit Problem in Celestial Mechanics, PhD thesis, ETH-Zürich (1989); see also “Analysis of resonances in the spin-orbit problem in Celestial Mechanics: the synchronous resonance (Part I),” Journal of Applied Mathematics and Physics (ZAMP), vol. 41, p.174-204 (1990).

[CC] A. Celletti and L. Chierchia, “KAM Stability and Celestial Mechanics,” Memoirs American Mathematical Society, vol. 187, no. 878 (2007).

[LG] U. Locatelli, A. Giorgilli, “Invariant Tori in the Secular Motions of the tTree-body Planetary Systems,” Celestial Mechanics and Dynamical Astronomy, vol. 78, 47-74 (2000).

[H] M. Hénon, “Explorationes numérique du problème restreint IV: Masses égales, orbites non périodique,” Bullettin Astronomique, vol. 3, 1, fasc. 2, 49-66 (1966).

[K] A.N. Kolmogorov, “On the conservation of conditionally periodic motions under small perturbation of the Hamiltonian,” Dokl. Akad. Nauk. SSR, vol. 98 527-530 (1954).

[M] J. Moser, “On invariant curves of area-preserving mappings of an annulus,” Nachr. Akad. Wiss. Göttingen, Math. Phys. Kl. II, vol. 1, 1-20 (1962).

Alessandra Celletti

Dipartimento di Matematica

Universita’ di Roma Tor Vergata

Italy