Geodesic
Topology of the configuration space, singularities of the potential, and polynomial integrals of equations of dynamics
Sbornik. Mathematics, Tome 207 (2016) no. 10, pp. 1435-1449 Cet article a éte moissonné depuis la source Math-Net.Ru

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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
Mots-clés : Poincaré condition. 
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V. V. Kozlov; D. V. Treschev. Topology of the configuration space, singularities of the potential, and polynomial integrals of equations of dynamics. Sbornik. Mathematics, Tome 207 (2016) no. 10, pp. 1435-1449. http://geodesic.mathdoc.fr/item/SM_2016_207_10_a4/

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