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Izv. RAN. Ser. Mat., 2017, Volume 81, Issue 4, Pages 3–19 (Mi izv8600)  

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

Topology, singularities and integrability in Hamiltonian systems with two degrees of freedom

S. V. Bolotinab, V. V. Kozlova

a Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
b University of Wisconsin-Madison

Abstract: We consider the problem of the existence of first integrals that are polynomial in momenta for Hamiltonian systems with two degrees of freedom on a fixed energy level (conditional Birkhoff integrals). It is assumed that the potential has several singular points. We show that in the presence of conditional polynomial integrals, the sum of degrees of the singularities does not exceed twice the Euler characteristic of the configuration space. The proof is based on introducing a complex structure on the configuration space and estimating the degree of the divisor corresponding to the leading term of the integral with respect to the momentum. We also prove that the topological entropy is positive under certain conditions.

Keywords: Hamiltonian system, integrability, singular point, regularization, Finsler metric, conformal structure.

Funding Agency Grant Number
Russian Academy of Sciences - Federal Agency for Scientific Organizations 0014-2015-0003
Russian Foundation for Basic Research 15-01-03747а
This paper was written with the support of the RAS Presidium programme ‘Mathematical problems of modern control theory’, project no. 0014-2015-0003 ‘Variational and extremal problems in dynamics’. The first author was partially supported by RFBR grant ‘Modern problems of classical dynamics’ (project no. 15-01-03747a).

DOI: https://doi.org/10.4213/im8600

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English version:
Izvestiya: Mathematics, 2017, 81:4, 671–687

Bibliographic databases:

UDC: 517.913+531.01
MSC: 37J30, 37K10, 70H05, 34C40
Received: 14.09.2016
Revised: 29.01.2017

Citation: S. V. Bolotin, V. V. Kozlov, “Topology, singularities and integrability in Hamiltonian systems with two degrees of freedom”, Izv. RAN. Ser. Mat., 81:4 (2017), 3–19; Izv. Math., 81:4 (2017), 671–687

Citation in format AMSBIB
\by S.~V.~Bolotin, V.~V.~Kozlov
\paper Topology, singularities and integrability in Hamiltonian systems with two degrees of freedom
\jour Izv. RAN. Ser. Mat.
\yr 2017
\vol 81
\issue 4
\pages 3--19
\jour Izv. Math.
\yr 2017
\vol 81
\issue 4
\pages 671--687

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    This publication is cited in the following articles:
    1. S. V. Bolotin, V. V. Kozlov, “Topological approach to the generalized $n$-centre problem”, Russian Math. Surveys, 72:3 (2017), 451–478  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    2. N. V. Denisova, “Polynomial integrals of mechanical systems on a torus with a singular potential”, Dokl. Phys., 62:8 (2017), 397–399  crossref  crossref  mathscinet  isi  elib  scopus
    3. I. S. Kharcheva, “Izoenergeticheskie mnogoobraziya integriruemykh bilyardnykh knizhek”, Vestn. Mosk. un-ta. Ser. 1. Matem., mekh., 2020, no. 4, 12–22  mathnet
    4. I. V. Volovich, “On Integrability of Dynamical Systems”, Proc. Steklov Inst. Math., 310 (2020), 70–77  mathnet  crossref  crossref  isi  elib
    5. 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  mathscinet  isi  scopus
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