Topological approach to the generalized $ n$-centre problem. (29th September 2017)

• Record Type: Journal Article
• Authors: Bolotin, S V; Kozlov, V V
• Is Part Of: Russian mathematical surveys. Volume 72:Number 3(2017)
• Page Start: 451
• Page End: 478
• Publication Date: 2017-09-29
• Subjects: Mathematics -- Soviet Union -- Periodicals; Mathematics -- Russia (Federation) -- Periodicals; Mathematics -- Periodicals; Mathematicians -- Soviet Union -- Periodicals; Mathematicians -- Russia (Federation) -- Periodicals; 510.5
• DOI: 10.1070/RM9779
• Languages: English
• ISSNs: 0036-0279

Abstract

Abstract: This paper considers a natural Hamiltonian system with two degrees of freedom and Hamiltonian$ H=\Vert p\Vert^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<A_{k+1}$ . It is proved that if $\displaystyle \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.