| Abstract: |
| In this talk we consider two attraction centres $c_1, c_2\in\mathbb{R}^2$ and study the dynamics of a particle $q\in\mathbb{R}^2\setminus\{c_1,c_2\}$ under the action of a singular potential function $U$ which has the following prescribed behaviour close to each centre $$ U(q)\sim \frac{m_j}{|q-c_j|^2},\qquad m_j>0,\ q\in\mathcal{U}(c_j). $$ We fix a positive value of the mechanical energy and we construct infinitely many periodic solutions in distinct homotopy classes, all avoiding collisions with the centres. We then investigate the existence of invariant sets that are topologically conjugate with the Bernoulli shift. Finally, we analyse the asymptotic behaviour of unbounded trajectories and discuss the existence of scattering solutions with prescribed topological behaviour with respect to the centres. This is a joint work with Stefano Baranzini and Alberto Boscaggin. |
|