Related papers: Symbolic dynamics: from the $N$-centre to the $(N+…
We consider the planar $N$-centre problem, with homogeneous potentials of degree $-\a<0$, $\a \in [1,2)$. We prove the existence of infinitely many collisions-free periodic solutions with negative and small energy, for any distribution of…
We study a class of singular dynamical systems which generalise the classical N-centre problem of Celestial Mechanics to the case in which the configuration space is a Riemannian surface. We investigate the existence of topological…
The planar $N$-centre problem describes the motion of a particle moving in the plane under the action of the force fields of $N$ fixed attractive centres: \[ \ddot{x}(t)=\sum_{j=1}^N\nabla V_j(x-c_j). \] In this paper we prove symbolic…
In the context of the Newtonian N-body problem, we prove the existence of a partially hyperbolic motion with prescribed positive energy and any initial collisionless configuration. Moreover, it is a free time minimizer of the respective…
This paper concerns the restricted 3-body problem. By applying topological methods we give a computer assisted proof of the existence of some classes of periodic orbits, the existence of symbolic dynamics and we give a rigorous lower…
We consider the dynamics of a Hamiltonian particle forced by a rapidly oscillating potential in $\dim$-dimensional space. As alternative to the established approach of averaging Hamiltonian dynamics by reformulating the system as…
In the Newtonian $n$-body problem for solutions with arbitrary energy, which start and end either at a total collision or a parabolic/hyperbolic infinity, we prove some basic results about their Morse and Maslov indices. Moreover for…
Consider the three-body problem with an attractive $1/r^2$ potential. Modulo symmetries, the dynamics of the bounded zero-angular momentum solutions is equivalent to a geodesic flow on the thrice-punctured sphere, or ``pair of pants''. The…
We investigate expansive solutions of the $N$-body problem in $\mathbb{R}^d$ ($d\ge2$) driven by homogeneous Newtonian potentials of degree $-\alpha$. We establish the existence of half-entire expansive motions with prescribed initial…
We explore variational approach to the finite-volume $N$-body problem. The general formalism for N non-relativistic spinless particles interacting with periodic pair-wise potentials yields N-body secular equations. The solutions depend on…
The Jacobi-Maupertuis metric provides a reformulation of the classical N-body problem as a geodesic flow on an energy-dependent metric space denoted $M_E$ where $E$ is the energy of the problem. We show that $M_E$ has finite diameter for $E…
Symbolic dynamics is applied to the planar three-body problem. Symbols are defined on the planar orbit when it experiences a syzygy crossing. If the body i is in the middle at the syzygy crossing and the vectorial area of the triangle made…
We prove a nonholonomic version of the classical Mauper\-tuis-Jacobi principle which transforms an autonomous mechanical nonholonomic problem, determined by a kinetic minus potential energy and a distribution, in a kinetic nonholonomic…
Consider the equal mass planar $4$-body problem with a potential corresponding to an inverse \textit{cube} force. The Jacobi-Maupertuis principle reparametrizes the dynamics as geodesics of a certain metric. We examine the curvature of this…
This paper we consider for the N-body problem with potential 1/r{\alpha} (0 < {\alpha} < 1) the existence of hyperbolic motions for any prescribed limit shape and any given initial configuration of the bodies. Here E is the Euclidean space…
We consider non-autonomous $N$-body-type problems with strong force type potentials at the origin and sub-quadratic growth at infinity, and using Ljusternik-Schnirelmann theory, we prove the existence of unbounded sequences of critical…
We study the dynamics of the planar circular restricted three-body problem in the context of a pseudo-Newtonian approximation. By using the Fodor-Hoenselaers-Perj\'es procedure, we perform an expansion in the mass potential of a static…
After the separation of the center-of-mass motion, a new privileged class of canonical Darboux bases is proposed for the non-relativistic N-body problem by exploiting a geometrical and group theoretical approach to the definition of {\it…
Consider the planar restricted $(N+1)$-body problem with trajectories of the $N(\ge 2)$ primaries forming a collision-free periodic solution of the $N$-body problem, for any positive energy $h$ and directions $\theta_{\pm} \in [0, 2\pi)$,…
We consider the classical three-dimensional motion in a potential which is the sum of $n$ attracting or repelling Coulombic potentials. Assuming a non-collinear configuration of the $n$ centres, we find a universal behaviour for all…