Topological approach to the generalized $n$-center problem

TL;DR

This paper extends topological criteria for chaos in Hamiltonian systems with singular potentials on closed surfaces, showing that certain singularity configurations guarantee chaotic dynamics regardless of potential specifics.

Contribution

It generalizes previous results by establishing topological conditions involving singularity types and counts that imply chaos in generalized $n$-center problems.

Findings

Systems with enough singularities of certain types exhibit chaotic invariant sets.

The criteria depend solely on topological properties, not on potential analytical details.

The approach applies to various surfaces, including the plane, using topological and regularization techniques.

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.