On November 30th 1896, Poincaré published a note entitled "On the periodic solutions and the least action principle" in the "Comptes rendus de l'Académie des Sciences". He proposed to find periodic solutions of the planar Three-Body Problem by minimizing the Lagrangian action among loops in the configuration space which satisfy given constraints (the constraints amount to fixing their homology class). For the Newtonian potential, proportional to the inverse of the distance, the "collision problem" prevented him from realizing his program; hence he replaced it by a "strong force potential" proportional to the inverse of the squared distance.
In the lecture, the nature of the difficulties met by Poincaré is explained and it is shown how, one century later, these have been partially resolved for the Newtonian potential, leading to the discovery of new remarkable families of periodic solutions of the planar or spatial $n$-body problem.

We are interested in studying the properties and perturbations of orbits around a central planet surrounded by a ring. The problem has been studied a long time ago by Laplace, Maxwell and others. Maxwell considered a ring composed of a number of discrete masses orbiting in a circular orbit. Gauss also derived the potential due to a solid circular ring and its derivation is reproduced in Kellog's textbook on Potential Theory. The potential can be evaluated in terms of a complete elliptic integral of the first kind. Computing the accelerations also requires a second kind elliptic integral. We have experimented with at least three different methods for computing the potential and its first partial derivatives: Gauss quadrature, the Carlson functions and the Arithmetico-Geometric mean. The standard formulation breaks down near the center of the ring which is an unstable equilibrium point but a linearization can be made near this point. We have also studied the efficiency of the Spherical Harmonic expansion of the ring potential. This expansion has only the even zonal terms and thus no tesserals.
In a preliminary study, we have looked at planar periodic orbits (and their stability), around a ring without a central body, both in the plane of the ring and the plane orthogonal to it. We find nearly a dozen types and families of periodic orbits. Several of these families seem to end with an orbit that collides with the ring. One of the goals of this preliminary study is to understand the effect of the singularity at the ring itself, (where the potential and the accelerations become infinite).

We show how, starting from the geometrical description of discrete Painlevé equations in terms of affine Weyl groups, one can generate new second-order systems. We use this approach to introduce a new definition of the discrete Painlevé equations which eschews the reference to continuous systems.

We describe an algorithm constructing an invariant KAM torus for a class of planetary systems, such that the mutual attractions, the eccentricities and the inclinations of the planets are small enough. By using computer algebra, we explicitly implement this algorithm for approximating a KAM torus for the problem of three bodies in a case similar to the Sun–Jupiter–Saturn system. We show that, by reducing the masses of the planets by a factor 10 and with a small displacement of the orbits, our semianalytical construction of the torus turns out to be successful.

We present a new method based on set oriented computations for the calculation of reaction rates in chemical systems. The method is demonstrated with the Rydberg atom, an example for which traditional Transition State Theory fails. Coupled with dynamical systems theory, the set oriented approach provides a global description of the dynamics. The main idea of the method is as follows. We construct a box covering of a Poincaré section under consideration, use the Poincaré first return time for the identification of those regions relevant for transport and then we apply an adaptation of recently developed techniques for the computation of transport rates ([12], [27]). The reaction rates in chemical systems are of great interest in chemistry, especially for realistic three and higher dimensional systems. Our approach is applied to the Rydberg atom in crossed electric and magnetic fields. Our methods are complementary to, but in common problems considered, agree with, the results of [14]. For the Rydberg atom, we consider the half and full scattering problems in both the 2- and the 3-degree of freedom systems. The ionization of such atoms is a system on which many experiments have been done and it serves to illustrate the elegance of our method.

We develop and analyze a single-degree-of-freedom model for multilegged locomotion in the horizontal (ground) plane. The model is a hybrid dynamical system incorporating rudimentary motoneuronal activation and agonist-antagonist Hill-type muscle pairs that drive a point mass body along a line. Composition of the smooth stance phase dynamics with various discrete lift-off and touch-down protocols result in Poincaré stride-to-stride maps, and our extreme simplification of the full dynamics permits relatively complete phase plane analysis of existence and stability of periodic gaits. We use these to perform parameter studies that provide guidelines for the construction of stable periodic orbits with appropriate force and velocity signatures.