Integrability, nonintegrability and chaotic motions for a system motivated by the Riemann ellipsoids problem

We consider a system obtained by coupling two Euler-Poinsot systems. The motivation to consider such a system can be traced back to the Riemann Ellipsoids problem. We deal with the problems of integrability and existence of region,of chaotic motions

Strong λ-lemma and second species periodic solutions of the 3-body problem

nuloQuite recently Bolotin and N. developed a variational approach to the existence of second species periodic solutions for the nonrestricted 3 body problem [2]. An important part of this paper is based on an analog of Shilnikov Lemma (also called strong λ- lemma) for hamiltonian systems possessing a non degenerate normally hyperbolic critical symplectic manifold. The complete proof of the Lemma was postponed to a future publication [3]: in this paper we anticipate a part of the results

On the Inversion of Lagrange-Dirichlet Theorem

We consider a Lagrangian differential system. The celebrated theorem of Lagrange-Dirichlet ensures that a stationary solution of this system is stable, provided that the corresponding critical point of the potential function is a proper {local} maximum

Stability Problems for the Euler Equation on the 2-Dimensional Torus

Stability Problems for the Euler Equation on the 2-Dimensional Toru

On the stationary motion of a self-gravitating toroidal stratum

We consider an incompressible fluid contained in a toroidal stratum and subject only to Newtonian self-attraction. Under the assumption of infinitesimal thickness of the stratum we show the existence of stationary motions during which the stratum is approximatly a round torus (with radiir, R and R >> r) that rotates around its axis and at the same time rolls on itself. Therefore each particle of the stratum describes an helix-like trajectory around the circumference of radius R.

Stationary motion of a self gravitating toroidal incompressible liquid layer

We consider an incompressible fluid contained in a toroidal stratum which is only subjected to Newtonian self-attraction. Under the assumption of infinitesimal tickness of the stratum we show the existence of stationary motions during which the stratum is approximatly a round torus (with radii r, R and R>>r) that rotates around its axis and at the same time rolls on itself. Therefore each particle of the stratum describes an helix-like trajectory around the circumference of radius R that connects the centers of the cross sections of the torus

Shilnikov Lemma for a nondegenerate critical manifold of a Hamiltonian system

We prove an analog of Shilnikov Lemma for a normally hyperbolic symplectic critical manifold $M\subset H^{-1}(0)$ of a Hamiltonian system. Using this result, trajectories with small energy $H=\mu>0$ 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 Poincar\'e second species solutions of the 3 body problem with 2 masses small of order $\mu$. As $\mu\to 0$, double collisions of small bodies correspond to a symplectic critical manifold of the regularized Hamiltonian system