Topological approach to the generalized n-centre problem

This paper considers a natural Hamiltonian system with two degrees of freedom and Hamiltonian H = parallel to p parallel to (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) similar to -dist(q, a(j))(-alpha j). Let A(k) = 2 - 2k(-1), k is an element of N, and let n(k) be the number of singular points with A(k) <= alpha(j) < A(k+ 1). It is proved that if Sigma(2 <= k <=infinity) n(k)A(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.