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)

Averaging the Planar Three-Body Problem in the Neighborhood of Double Inner Collisions

Arnold's theorem on the planetary problem states that, assuming that
 the masses of $n$ planets are small enough, there exists in the
 phase space a set of initial conditions of positive Lebesgue
 measure, leading to quasiperiodic motions with $3n-1$
 frequencies. Arnold's initial proof is complete only for the plane
 $2$-planet problem. Arnold had missed a resonance later discovered
 by Herman. The first complete proof, by Herman-Fejoz, relies on the
 weak non-degeneracy condition of Arnold-Pyartli. A second proof, by
 Chierchia-Pinzari, is closer to Arnold's initial idea and shows the
 strong non-degeneracy of the problem after suitable reduction by
 (part of) the symmetry of rotation. We review and compare these
 proofs. In an appendix, we define the Poincare coordinates and prove
 their symplectic nature through the shortest possible computation.

/pdf/on-arnold-s-theorem-on-the-stability-of-the-solar-system-qvyyssgymr.pdf

On "Arnold's theorem" on the stability of the solar system

We prove a new invariant torus theorem, for α-Gevrey smooth Hamiltonian systems , under an arithmetic assumption which we call the α-Bruno-Russmann condition , and which reduces to the classical Bruno-Russmann condition in the analytic category. Our proof is direct in the sense that, for analytic Hamiltonians, we avoid the use of complex extensions and, for non-analytic Hamiltonians, we do not use analytic approximation nor smoothing operators. Following Bessi, we also show that if a slightly weaker arithmetic condition is not satisfied, the invariant torus may be destroyed. Crucial to this work are new functional estimates in the Gevrey class.

/pdf/kam-a-gevrey-regularity-and-the-a-bruno-russmann-condition-58ddlwftf1.pdf

KAM, α -Gevrey regularity and the α -Bruno-Rüssmann condition

Consider the three-body problem, in the regime where one body revolves far away around the other two, in space, the masses of the bodies being arbitrary but fixed; in this regime, there are no resonances in mean motions. The so-called secular dynamics governs the slow evolution of the Keplerian ellipses. We show that it contains a horseshoe and all the chaotic dynamics which goes along with it, corresponding to motions along which the eccentricity of the inner ellipse undergoes large, random excursions. The proof goes through the surprisingly explicit computation of the homoclinic solution of the first order secular system, its complex singularities and the Melnikov potential.

/pdf/secular-instability-in-the-three-body-problem-363xko80xc.pdf

Secular Instability in the Three-Body Problem

A variant of Kolmogorov’s initial proof is given, in terms of a group of symplectic transformations and of an elementary fixed point theorem.

Jacques Féjoz

Papers

Averaging the Planar Three-Body Problem in the Neighborhood of Double Inner Collisions

On "Arnold's theorem" on the stability of the solar system

KAM, α -Gevrey regularity and the α -Bruno-Rüssmann condition

Secular Instability in the Three-Body Problem

A proof of the invariant torus theorem of Kolmogorov