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Mat. Sb., 2016, Volume 207, Number 10, Pages 80–95 (Mi msb8786)  

This article is cited in 7 scientific papers (total in 7 papers)

Topology of the configuration space, singularities of the potential, and polynomial integrals of equations of dynamics

V. V. Kozlov, D. V. Treschev

Steklov Mathematical Institute of Russian Academy of Sciences, Moscow

Abstract: For integrable systems with two degrees of freedom there are well-known inequalities connecting the Euler characteristic of the configuration space (as a closed two-dimensional surface) with the number of singular points of Newtonian type of the potential energy. On the other hand, there are results on conditions for ergodicity of systems on a two-dimensional torus with short-range potential depending only on the distance from an attracting or repelling centre. In the present paper we consider the problem of conditions for the existence of nontrivial first integrals that are polynomial in the momenta of the problem of motion of a particle on a multi-dimensional Euclidean torus in a force field whose potential has singularity points. These conditions depend only on the order of the singularity, and in the two-dimensional case they are satisfied by potentials with singularities of Newtonian type.
Bibliography: 13 titles.

Keywords: polynomial integrals, potentials with singularities, order of singularity, Poincaré condition.

Funding Agency Grant Number
Russian Science Foundation 14-50-00005
This research was supported by the Russian Science Foundation (project no. 14-50-00005).


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English version:
Sbornik: Mathematics, 2016, 207:10, 1435–1449

Bibliographic databases:

UDC: 517.913
MSC: Primary 70G40; Secondary 37D50, 37J35, 70G10, 70H06, 70H07
Received: 14.06.2016 and 18.08.2016

Citation: V. V. Kozlov, D. V. Treschev, “Topology of the configuration space, singularities of the potential, and polynomial integrals of equations of dynamics”, Mat. Sb., 207:10 (2016), 80–95; Sb. Math., 207:10 (2016), 1435–1449

Citation in format AMSBIB
\by V.~V.~Kozlov, D.~V.~Treschev
\paper Topology of the configuration space, singularities of the~potential, and polynomial integrals of equations of dynamics
\jour Mat. Sb.
\yr 2016
\vol 207
\issue 10
\pages 80--95
\jour Sb. Math.
\yr 2016
\vol 207
\issue 10
\pages 1435--1449

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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. 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
    3. 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
    4. L. V. Lokutsievskiy, Yu. L. Sachkov, “Liouville integrability of sub-Riemannian problems on Carnot groups of step 4 or greater”, Sb. Math., 209:5 (2018), 672–713  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    5. I. V. Volovich, V. Zh. Sakbaev, “On quantum dynamics on $C^*$-algebras”, Proc. Steklov Inst. Math., 301 (2018), 25–38  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    6. I. V. Volovich, “On Integrability of Dynamical Systems”, Proc. Steklov Inst. Math., 310 (2020), 70–77  mathnet  crossref  crossref  isi  elib
    7. 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
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