Propositum est mihi, Lector, hoc libello demonstrare, quod Creator Optimus Maximus, in creatione mundi huius mobilis, et dispositione coelorum, ad illa quinque regularia corpora, inde a Pythagora et Platone, ad nos utque, celebratissima respexerit, atque ad illorum naturam coelorum numerum, proportiones, et motuum rationem accommodaverit. (J. Kepler, Myst. Cosm. [20])Abstract We prove the existence of a number of smooth periodic motions u * of the classical Newtonian N -body problem which, up to a relabeling of the N particles, are invariant under the rotation group R of one of the five Platonic polyhedra. The number N coincides with the order |R| of R and the particles have all the same mass. Our approach is variational and u * is a minimizer of the Lagrangean action A on a suitable subset K of the H 1 T -periodic maps u : R → R 3N . The set K is a cone and is determined by imposing to u both topological and symmetry constraints which are defined in terms of the rotation group R. There exist infinitely many such cones K, all with the property that A| K is coercive. For a certain number of them, using level estimates and local deformations, we show that minimizers are free of collisions and therefore classical solutions of the N -body problem with a rich geometric-kinematic structure. 2

We consider the plane 3 body problem with 2 of the masses small. Periodic solutions with near collisions of small bodies were named by Poincaré second species periodic solutions. Such solutions shadow chains of collision orbits of 2 uncoupled Kepler problems. Poincaré only sketched the proof of the existence of second species solutions. Rigorous proofs appeared much later and only for the restricted 3 body problem. We develop a variational approach to the existence of second species periodic solutions for the nonrestricted 3 body problem. As an application, we give a rigorous proof of the existence of a class of second species solutions.

We provide a result of non-analytic integrability of the so-called J 2-problem. Precisely by using the Lerman theorem we are able to prove the existence of a region of the phase space, where the dynamical system exhibits chaotic motions

We show that the n-center problem in ℝ3 has positive topological entropy for n ≥ 3. The proof is based on global regularization of singularities and the results of Gromov and Paternain on the topological entropy of geodesic flows. The n-center problem in S3 is also studied

We prove an analog of Shilnikov Lemma for a normally hyperbolic symplectic critical manifold M ⊂ H −1 (0) of a Hamiltonian system. Using this result, trajectories with small energy H = µ > 0 shadowing chains of homoclinic orbits to M are represented as extremals of a discrete variational problem, and their existence is proved. This paper is motivated by applications to the Poincaré second species solutions of the 3 body problem with 2 masses small of order µ. As µ → 0, double collisions of small bodies correspond to a symplectic critical manifold of the regularized Hamiltonian system. * Supported by the Programme "Dynamical Systems and Control Theory" of RAS and RFBR grants #12-01-00441 and #13-01-12462.Let Dφ t (z) = e tA(z) be the linearized flow. Denote bywhere the eigenvalues of A ± (z) = ∓A(z)| E ± z are positive. Thus E + z is the stable subspace, and E − z the unstable subspace. The quadratic part of the Hamiltonian is 1 2( 1.2)The stable and unstable manifolds 1of M have dimension k + 2m and T z W ± = T z M ⊕ E ± z for any z ∈ M . It is well known (see e.g. [11]) that W ± (z) are isotropic: ω| W ± (z) = 0, and W ± are coisotropic: for any a ∈ W ± (z), we have T ⊥ a W ± = T a W ± (z). Thus W ± (z) form a smooth isotropic foliation of W ± . Define projections π ± : W ± → M by π ± (x) = z if x ∈ W ± (z):Remark 1.1. Following [11], we call F the scattering map. However, our case is different from [11] because the manifold M is critical. In particular, there is no straightforward cross section for the flow near M . The scattering map is also called the homoclinic map. In the applications to Celestial Mechanics [5,8], we call F the collision map.

We consider the motion of a particle in a plane under the gravitational action of 3 fixed centers (the 3-center planar problem). As it is well known (Vestnik Moskov. Univ. Ser. 1 Matem. Mekh 6 (1984) 65; Prik1. Matem. i Mekhan. 48 (1984) 356; Classical Planar Scattering by Coulombic Potentials, Lecture Notes in Physics, Springer, Berlin, 1992) on the non-negative level sets of the energy E there do not exist non-constant analytic first integrals, and moreover the system has chaotic trajectories. These results were proved by variational methods. Here we investigate the problem in the domain of small negative values of E. Moreover, we assume that one of the centers is far away from the other two. Then we get a two-parameter singular perturbation of an integrable dynamical system: the 2-center problem on the zero-energy level. The main problem we deal with is to prove that the Poincare-Melnikov theory applies in the limit case E --> 0. (C) 2003 Elsevier Science (USA). All rights reserved

A concept of total stability for continuous or discrete dynamical systems and a generalized definition of bifurcation are given: it is possible to show the link between an abrupt change of the asymptotic behaviour of a family of flows and the arising of new invariant sets, with determined asymptotic properties. The theoretical results are a contribution to the study of the behaviour of flows near an invariant compact set. They are obtained by means of an extension of Liapunov's direct method. © 1975 Fondazione Annali di Matematica Pura ed Applicata

We introduce a dynamical system strictly related to fluid mechanics and similar to the classical N point vortex system. In the first part we analyze the qualitative behavior of the time evolution and in particular we show the properties of collapse and chaoticity. In the second part of the paper we investigate the relation of the dynamical system with a system of N concentrated large enough smoke rings in an incompressible and inviscid fluid, with axial symmetry and without swirl. We prove the rigorous connection between the two models at time zero for any N . The extension of the same result to any time is obtained only for a smoke ring alone, while for the general case it is just a matter of conjecture.