Topological approach to the generalized $n$-center problem

Abstract

We consider 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 noncompact $M$ 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 energy levels $H=h>\sup V$. We generalize this result to the case when the potential energy has several singular points $a_j$ of type $V(q)\sim -d(q,a_j)^{-\alpha_j}$. Let $A_k=2-2k^{-1}$, $k=2,3,\dots$, and let $n_k$ be the number of singular points with $A_k\le \alpha_j<A_{k+1}$. We prove that if $$ \sum_{2\le k\le\infty}n_kA_k>2\chi(M), $$ then the system has a compact chaotic invariant set of noncollision trajectories on any energy level $H=h>\sup V$. This result is purely topological: no analytical properties of the potential, except the presence of singularities, are involved. The proofs are based on the generalized Levi-Civita regularization and elementary topology of coverings. As an example, the plane $n$ center problem is considered.