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

Part I. Examples and Fundamental Concepts Introduction 1. First examples 2. Equivalence, classification, and invariants 3. Principle classes of asymptotic invariants 4. Statistical behavior of the orbits and introduction to ergodic theory 5. Smooth invariant measures and more examples Part II. Local Analysis and Orbit Growth 6. Local hyperbolic theory and its applications 7. Transversality and genericity 8. Orbit growth arising from topology 9. Variational aspects of dynamics Part III. Low-Dimensional Phenomena 10. Introduction: What is low dimensional dynamics 11. Homeomorphisms of the circle 12. Circle diffeomorphisms 13. Twist maps 14. Flows on surfaces and related dynamical systems 15. Continuous maps of the interval 16. Smooth maps of the interval Part IV. Hyperbolic Dynamical Systems 17. Survey of examples 18. Topological properties of hyperbolic sets 19. Metric structure of hyperbolic sets 20. Equilibrium states and smooth invariant measures Part V. Sopplement and Appendix 21. Dynamical systems with nonuniformly hyperbolic behavior Anatole Katok and Leonardo Mendoza.

Introduction to the Modern Theory of Dynamical Systems: EQUILIBRIUM STATES AND SMOOTH INVARIANT MEASURES

Using a variational method, we exhibit a surprisingly simple periodic orbit for the newtonian problem of three equal masses in the plane. The orbit has zero angular momentum and a very rich symmetry pattern. Its most surprising feature is that the three bodies chase each other around a flxed eight-shaped curve. Setting aside collinear motions, the only other known motion along a flxed curve in the inertial plane is the \Lagrange relative equilibrium" in which the three bodies form a rigid equilateral triangle which rotates at constant angular velocity within its circumscribing circle. Our orbit visits in turns every \Euler conflguration" in which one of the bodies sits at the midpoint of the segment deflned by the other two (Figure 1). Numerical computations

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A remarkable periodic solution of the three-body problem in the case of equal masses

Using a variational method, we exhibit a surprisingly simple periodic orbit for the newtonian problem of three equal masses in the plane. The orbit has zero angular momentum and a very rich symmetry pattern. Its most surprising feature is that the three bodies chase each other around a fixed eight-shaped curve. Setting aside collinear motions, the only other known motion along a fixed curve in the inertial plane is the ``Lagrange relative equilibrium" in which the three bodies form a rigid equilateral triangle which rotates at constant angular velocity within its circumscribing circle. Our orbit visits in turns every ``Euler configuration" in which one of the bodies sits at the midpoint of the segment defined by the other two (Figure 1). Numerical computations by Carles Simo, to be published elsewhere, indicate that the orbit is ``stable" (i.e. completely elliptic with torsion). Moreover, they show that the moment of inertia I(t) with respect to the center of mass and the potential U(t) as functions of time are almost constant.

We show that the minimization of the Lagrangian action functional on suitable classes of symmetric loops yields collisionless periodic orbits of the n-body problem, provided that some simple conditions on the symmetry group are satisfied. More precisely, we give a fairly general condition on symmetry groups G of the loop space Λ for the n-body problem (with potential of homogeneous degree -α, with α>0) which ensures that the restriction of the Lagrangian action $\mathcal{A}$
 to the space Λ
 G
 of G-equivariant loops is coercive and its minimizers are collisionless, without any strong force assumption. In proving that local minima of Λ
 G
 are free of collisions we develop an averaging technique based on Marchal’s idea of replacing some of the point masses with suitable shapes (see [10]). As an application, several new orbits can be found with some appropriate choice of G. Furthermore, the result can be used to give a simplified and unitary proof of the existence of many already known minimizing periodic orbits.

http://www.matapp.unimib.it/~ferrario/OLDHTML/papers/ferter.pdf

On the existence of collisionless equivariant minimizers for the classical n -body problem

Le problème des n corps et les distances mutuelles

On etudie les proprietes dynamiques de certaines familles a 2 parametres generiques de germes de diffeomorphismes de R 2

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Bifurcations de points fixes elliptiques - I. Courbes invariantes

A “simple choreography” for an N-body problem is a periodic solution in which all N masses trace the same curve without colliding. We shall require all masses to be equal and the phase shift between consecutive bodies to be constant. The first 3-body choreography for the Newtonian potential, after Lagrange’s equilateral solution, was proved to exist by Chenciner and Montgomery in December 1999 (Chenciner and Montgomery [2000]). In this paper we prove the existence of planar N-body simple choreographies with arbitrary complexity and/or symmetry, and any number N of masses, provided the potential is of strong force type (behaving like 1/r a , a≥2 as r→0). The existence of simple choreographies for the Newtonian potential is harder to prove, and we fall short of this goal. Instead, we present the results of a numerical study of the simple Newtonian choreographies, and of the evolution with respect to a of some simple choreographies generated by the potentials 1/r a , focusing on the fate of some simple choreographies guaranteed to exist for a≥2 which disappear as a tends to 1.

Simple Choreographic Motions of N Bodies: A Preliminary Study

Formes normales. Orbites bien ordonnees. Estimation a priori sur C. Estimations lipschitziennes sur les orbites bien ordonnees. Elimination des problemes de bord. Existence d'ensembles invariants d'Aubry-Mather. Ensembles translates d'Aubry-Mather

Bifurcations de points fixes elliptiques. II. Orbites périodiques et ensembles de Cantor invariants.

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Bifurcations de points fixes elliptiques - III. Orbites périodiques de « petites » périodes et élimination résonnante des couples de courbes invariantes

Alain Chenciner

Papers

Le problème des n corps et les distances mutuelles

Bifurcations de points fixes elliptiques - I. Courbes invariantes

Simple Choreographic Motions of N Bodies: A Preliminary Study

Bifurcations de points fixes elliptiques. II. Orbites périodiques et ensembles de Cantor invariants.

Bifurcations de points fixes elliptiques - III. Orbites périodiques de « petites » périodes et élimination résonnante des couples de courbes invariantes