Topological approach to the generalized $n$-centre problem

S. V. Bolotin ; V. V. Kozlov

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 72 (2017) no. 3, pp. 451-478

Résumé

This paper considers a natural Hamiltonian system with two degrees of freedom and Hamiltonian $H=\|p\|^2/2+V(q)$. The configuration space $M$ is a closed surface (for non-compact $M$ certain conditions at infinity are required). It is well known that if the potential energy $V$ has $n>2\chi(M)$ Newtonian singularities, then the system is not integrable and has positive topological entropy on the energy level $H=h>\sup V$. This result is generalized here to the case when the potential energy has several singular points $a_j$ of type $V(q)\sim {-}\operatorname{dist}(q,a_j)^{-\alpha_j}$. Let $A_k=2-2k^{-1}$, $k\in\mathbb{N}$, and let $n_k$ be the number of singular points with $A_k\leqslant \alpha_j$. It is proved that if $$ \sum_{2\leqslant k\leqslant\infty}n_kA_k>2\chi(M), $$ then the system has a compact chaotic invariant set of collision-free trajectories on any energy level $H=h>\sup V$. This result is purely topological: no analytical properties of the potential energy are used except the presence of singularities. The proofs are based on the generalized Levi-Civita regularization and elementary topology of coverings. As an example, the plane $n$-centre problem is considered. Bibliography: 29 titles.

Détail

Keywords: Hamiltonian system, integrability, singular point, degree of singular point, Levi-Civita regularization, Finsler metric, covering, collision-free trajectory, chaotic invariant set, metric space, Jacobi metric.