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

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).

DOI: https://doi.org/10.4213/sm8786

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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

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
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• http://mi.mathnet.ru/eng/msb8786
• https://doi.org/10.4213/sm8786
• http://mi.mathnet.ru/eng/msb/v207/i10/p80

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Citing articles on Google Scholar: Russian citations, English citations
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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
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
3. N. V. Denisova, “Polynomial integrals of mechanical systems on a torus with a singular potential”, Dokl. Phys., 62:8 (2017), 397–399
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
5. I. V. Volovich, V. Zh. Sakbaev, “On quantum dynamics on $C^*$-algebras”, Proc. Steklov Inst. Math., 301 (2018), 25–38
6. I. V. Volovich, “On Integrability of Dynamical Systems”, Proc. Steklov Inst. Math., 310 (2020), 70–77
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
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