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This article is cited in 10 scientific papers (total in 10 papers)
Topological approach to the generalized $n$-centre problem
S. V. Bolotin, V. V. Kozlov Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
Abstract:
This paper considers 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 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
$$
\sum_{2\leqslant k\leqslant\infty}n_kA_k>2\chi(M),
$$
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.
Keywords:
Hamiltonian system, integrability, singular point, degree of singular point, Levi-Civita regularization, Finsler metric, covering, collision-free trajectory, chaotic invariant set, metric space, Jacobi metric.
Received: 25.04.2017
Citation:
S. V. Bolotin, V. V. Kozlov, “Topological approach to the generalized $n$-centre problem”, Russian Math. Surveys, 72:3 (2017), 451–478
Linking options:
https://www.mathnet.ru/eng/rm9779https://doi.org/10.1070/RM9779 https://www.mathnet.ru/eng/rm/v72/i3/p65
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