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)

Quasiperiodic motions in the planar three-body problem

We show that three families of relative periodic solutions bifurcate out of the Eight solution of the equal-mass three-body problem: the planar Henon family, the spatial Marchal P12 family and a new spatial family. The Eight, considered as a spatial curve, is invariant under the action of the 24-element group D6 × Z2. The three families correspond to symmetry breakings where the invariance group becomes isomorphic to D6, the three D6s being embedded in the larger group in different ways. The proof of the existence of these three families relies on writing down the action integral in a rotating frame, viewing the angular velocity of the frame as a parameter, exploiting the invariance of the action under a group action which acts on the angular velocities as well as the curves and, finally, checking numerically the non-degeneracy of the Eight. Pictures and numerical evidence of the three families are presented at the end.

Rotating Eights: I. The three Γi families

The simplest solutions of the N-body problem --symmetric relative equilibria-- are shown to be organizing centers from which stem some recently studied classes of periodic solutions. We focus on the relative equilibrium of the equal-mass regular N-gon, assumed horizontal, and study the families of Lyapunov quasi-periodic solutions bifurcating from them in the vertical direction. The proof of the local existence of such solutions relies on the fact that the restriction to the corresponding directions of the quadratic part of the energy is positive definite. We then discuss the possibility of continuing the families globally as action minimizers under symmetry constraints by using the fact that, in rotating frames where they become periodic, these solutions are highly symmetric. The paradigmatic examples are the ``Eight'' families for an odd number of bodies and the ``Hip-Hop'' families for an even number. We argue that it is precisely for these two families that global minimization may be used. We also study the relation with the regular N-gon, of the so-called ``chain'' choreographies (see C. Simo, New families of Solutions in N-Body Problems, Progr. Math. 201, 2001): here, only a local minimization property is true (except for N=3) and moreover the parity plays a deciding role, in particular through the value of the angular momentum.

/pdf/unchained-polygons-and-the-n-body-problem-599vb6rhya.pdf

Unchained polygons and the N -body problem

We study the dynamics of the restricted planar three-body problem near mean motion resonances, i.e. a resonance involving the Keplerian periods of the two lighter bodies revolving around the most massive one. This problem is often used to model Sun--Jupiter--asteroid systems. For the primaries (Sun and Jupiter), we pick a realistic mass ratio $\mu=10^{-3}$ and a small eccentricity $e_0>0$. The main result is a construction of a variety of non local diffusing orbits which show a drastic change of the osculating (instant) eccentricity of the asteroid, while the osculating semi major axis is kept almost constant. The proof relies on the careful analysis of the circular problem, which has a hyperbolic structure, but for which diffusion is prevented by KAM tori. We verify certain non-degeneracy conditions numerically. Based on the work of Treschev, it is natural to conjecture that diffusion time for this problem is $\sim \frac{-\ln (\mu e_0)}{\mu^{3/2} e_0}$. We expect our instability mechanism to apply to realistic values of $e_0$ and we give heuristic arguments in its favor. If so, the applicability of Nekhoroshev theory to the three-body problem as well as the long time stability become questionable. It is well known that, in the Asteroid Belt, located between the orbits of Mars and Jupiter, the distribution of asteroids has the so-called Kirkwood gaps exactly at mean motion resonances of low order. Our mechanism gives a possible explanation of their existence. To relate the existence of Kirkwood gaps with Arnold diffusion, we state a conjecture on its existence for a typical $\eps$-perturbation of the product of a pendulum and a rotator. Namely, we predict that a positive conditional measure of initial conditions concentrated in the main resonance exhibits Arnold diffusion on time scales $\frac{- \ln \eps}{\eps^{2}}$.

https://www.ems-ph.org/journals/show_pdf.php?issn=1435-9855&vol=18&iss=10&rank=3

Jacques Féjoz

Papers

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

Quasiperiodic motions in the planar three-body problem

Rotating Eights: I. The three Γi families

Unchained polygons and the N -body problem

Kirkwood gaps and diffusion along mean motion resonances in the restricted planar three-body problem