Uspekhi Matematicheskikh Nauk
General information
Latest issue
Impact factor
License agreement
Submit a manuscript

Search papers
Search references

Latest issue
Current issues
Archive issues
What is RSS

Uspekhi Mat. Nauk:

Personal entry:
Save password
Forgotten password?

Uspekhi Mat. Nauk, 2017, Volume 72, Issue 3(435), Pages 65–96 (Mi umn9779)  

This article is cited in 5 scientific papers (total in 5 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.

Funding Agency Grant Number
Russian Science Foundation 14-50-00005
This work was supported by the Russian Science Foundation under grant 14-50-00005.


Full text: PDF file (722 kB)
References: PDF file   HTML file

English version:
Russian Mathematical Surveys, 2017, 72:3, 451–478

Bibliographic databases:

UDC: 517.913+531.01
MSC: Primary 70F10; Secondary 37N05, 70G40
Received: 25.04.2017

Citation: S. V. Bolotin, V. V. Kozlov, “Topological approach to the generalized $n$-centre problem”, Uspekhi Mat. Nauk, 72:3(435) (2017), 65–96; Russian Math. Surveys, 72:3 (2017), 451–478

Citation in format AMSBIB
\by S.~V.~Bolotin, V.~V.~Kozlov
\paper Topological approach to the generalized $n$-centre problem
\jour Uspekhi Mat. Nauk
\yr 2017
\vol 72
\issue 3(435)
\pages 65--96
\jour Russian Math. Surveys
\yr 2017
\vol 72
\issue 3
\pages 451--478

Linking options:

    SHARE: FaceBook Twitter Livejournal

    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. S. V. Bolotin, V. V. Kozlov, “Topology, singularities and integrability in Hamiltonian systems with two degrees of freedom”, Izv. Math., 81:4 (2017), 671–687  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    2. A. Boscaggin, W. Dambrosio, D. Papini, “Parabolic solutions for the planar $N$-centre problem: multiplicity and scattering”, Ann. Mat. Pura Appl. (4), 197:3 (2018), 869–882  crossref  mathscinet  zmath  isi  scopus
    3. A. Boscaggin, A. Bottois, W. Dambrosio, “The spatial $N$ -centre problem: scattering at positive energies”, Calc. Var. Partial Differential Equations, 57:5 (2018), 118, 23 pp.  crossref  mathscinet  isi  scopus
    4. K.-Ch. Chen, G. Yu, “Variational construction for heteroclinic orbits of the n-center problem”, Calc. Var. Partial Differ. Equ., 59:1 (2020), 4  crossref  isi
    5. N. V. Denisova, “On Momentum-Polynomial Integrals of a Reversible Hamiltonian System of a Certain Form”, Proc. Steklov Inst. Math., 310 (2020), 131–136  mathnet  crossref  crossref  mathscinet  isi  elib
  • Успехи математических наук Russian Mathematical Surveys
    Number of views:
    This page:541
    Full text:56
    First page:36

    Contact us:
     Terms of Use  Registration to the website  Logotypes © Steklov Mathematical Institute RAS, 2021