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Topological approach to the generalized $n$-center problem | | Sergey Bolotin
; Valery Kozlov
; | | Date: |
12 May 2017 | | 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 $n>2chi(M)$ 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 $A_kle alpha_j<A_{k+1}$. We prove that if
$$ sum_{2le kleinfty}n_kA_k>2chi(M), $$ 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. | | Source: | arXiv, 1705.4671 | | Services: | Forum | Review | PDF | Favorites | |
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