Initial-value ordinary differential equation solution via variable order Adams method (SIVA/DIVA) package is collection of subroutines for solution of nonstiff ordinary differential equations. There are versions for single-precision and double-precision arithmetic. Requires fewer evaluations of derivatives than other variable-order Adams predictor/ corrector methods. Option for direct integration of second-order equations makes integration of trajectory problems significantly more efficient. Written in FORTRAN 77.

Solving Ordinary Differential Equations

The theory of pseudo differential operators, discussed in § 1, is well suited for investigating various problems connected with elliptic differential equations. However, this theory fails to be adequate for studying equations of hyperbolic type, and one is then forced to examine a wider class of operators, the so-called Fourier integral operators (Egorov [1975], Hormander [1968, 1971, 1983, 1985], Kumano-go [1982], Shubin [1978], Taylor [1981], Treves [1980]).

Fourier Integral Operators

FROM the beginning of civilization one need must make itself felt, namely, for some form of calendar. In turn, this gives rise to the primitive study of astronomy, involving some knowledge of the sun and the moon. When the planets are added in the next stage, the astronomer needs an ephemeris or almanac ; the modern Nautical Almanac, with its perfection within narrow limits, is the final outcome. There have been many stages on the road, and some apparent interruptions. But the urge has always been the same, essentially a practical one. The Analytical Foundations of Celestial Mechanics By Aurel Wintner. (Princeton Mathematical Series, No. 5.) Pp. xii + 448. (Princeton, N.J.: Princeton University Press; London: Oxford University Press, 1941.) 36s. net.

The Analytical Foundations of Celestial Mechanics

The problem is to determine the possible motions of three point masses \$m_1\\ ,\$ \$m_2\\ ,\$ and \$m_3\\ ,\$ which attract each other according to Newton's law of inverse squares. It started with the perturbative studies of Newton himself on the inequalities of the lunar motion[1]. In the 1740s it was constituted as the search for solutions (or at least approximate solutions) of a system of ordinary differential equations by the works of Euler, Clairaut and d'Alembert (with in particular the explanation by Clairaut of the motion of the lunar apogee). Much developed by Lagrange, Laplace and their followers, the mathematical theory entered a new era at the end of the 19th century with the works of Poincaré and since the 1950s with the development of computers. While the two-body problem is integrable and its solutions completely understood (see [2],[AKN],[Al],[BP]), solutions of the three-body problem may be of an arbitrary complexity and are very far from being completely understood.

Three body problem

V. I. Arnold (Petits denominateurs et problemes de stabilite du mouvement en mecanique classique et en mecanique celeste. Usp. Mat. Nauk. 18 (1963), 91–192 (en russe)) a affirme et partiellement demontre que, pour le modele newtonien du Systeme solaire a $n\geq 2$ planetes dans l'espace, si la masse des planetes est suffisamment petite par rapport a celle du Soleil, il existe, dans l'espace des phases au voisinage des mouvements kepleriens circulaires coplanaires, un sous-ensemble de mesure de Lebesgue strictement positive de conditions initiales conduisant a des mouvements quasiperiodiques a 3n - 1 frequences. Cet article detaille la demonstration que M. R. Herman a exposee de ce theoreme (Demonstration d'un theoreme de V. I. Arnold, Seminaire de Systemes Dynamiques et manuscrits, 1998).

Demonstration du `theoreme d'Arnold' sur la stabilite du systeme planetaire (d'apres Herman)

Non-integrability of the minimum-time Kepler problem

A longstanding belief has been that the semimajor axes, in the Newtonian planetary problem, are stable. In the course of the XIX century, Laplace, Lagrange and others gave stronger and stronger arguments in this direction, thus culminating in what has commonly been referred to as the first Laplace-Lagrange stability theorem. In the problem with 3 planets, we prove the existence of orbits along which the semimajor axis of the outer planet undergoes large random variations thus disproving the theorem of Laplace-Lagrange. The time of instability varies as a negative power of the masses of the planets. The orbits we have found fall outside the scope of the theory of Nekhoroshev-Niederman because they are not confined by the conservation of angular momentum and because the Hamiltonian is not (uniformly) convex with respect to the Keplerian actions.

A counterexample to the theorem of Laplace-Lagrange on the stability of semimajor axes

We study invariant manifolds of conformal symplectic dynamical systems on a symplectic manifold (M, $\omega$) of dimension $\ge$4. This class of systems is the 1-dimensional extension of symplectic dynamical systems for which the symplectic form is transformed colinearly to itself. In this context, we first examine how the $\omega$-isotropy of an invariant manifold N relates to the entropy of the dynamics it carries. Central to our study is Yomdin's inequality, and a refinement obtained using that the local entropies have no effect transversally to the characteristic foliation of N. When (M, $\omega$) is exact and N is isotropic, we also show that N must be exact for some choice of the primitive of $\omega$, under the condition that the dynamics acts trivially on the cohomology of degree 1 of N. The conclusion partially extends to the case when N has a relatively compact one-sided orbit. We eventually prove the uniqueness of invariant submanifolds N when M is a cotangent bundle, provided that the dynamics is isotopic to the identity among Hamiltonian diffeomorphisms. In the case of the cotangent bundle of the torus, a theorem of Shelukhin allows us to conclude that N is unique even among submanifolds with compact orbits.

Invariant submanifolds of conformal symplectic dynamics

EOR ` EME SUR LES ACTIONS DE GROUPES DE DIMENSION INFINIE par

Je dresse une liste d'erreurs de la Demonstration du "theoreme d'Arnold'' sur la stabilite du systeme planetaire [Ergodic Theory and Dynamical Systems 24:5 (2004) 1521-1582] ; aucune ne remet la demonstration en jeu. Quelques precisions et complements sont apportes.

Jacques Féjoz

Papers

Non-integrability of the minimum-time Kepler problem

A counterexample to the theorem of Laplace-Lagrange on the stability of semimajor axes

Invariant submanifolds of conformal symplectic dynamics

EOR ` EME SUR LES ACTIONS DE GROUPES DE DIMENSION INFINIE par

Errata de la Démonstration du "théorème d'Arnold'' et addendum