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
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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, Poincaré condition.
@article{SM_2016_207_10_a4,
author = {V. V. Kozlov and D. V. Treschev},
title = {Topology of the configuration space, singularities of the~potential, and polynomial integrals of equations of dynamics},
journal = {Sbornik. Mathematics},
pages = {1435--1449},
publisher = {mathdoc},
volume = {207},
number = {10},
year = {2016},
language = {en},
url = {http://geodesic.mathdoc.fr/item/SM_2016_207_10_a4/}
}
TY - JOUR AU - V. V. Kozlov AU - D. V. Treschev TI - Topology of the configuration space, singularities of the~potential, and polynomial integrals of equations of dynamics JO - Sbornik. Mathematics PY - 2016 SP - 1435 EP - 1449 VL - 207 IS - 10 PB - mathdoc UR - http://geodesic.mathdoc.fr/item/SM_2016_207_10_a4/ LA - en ID - SM_2016_207_10_a4 ER -
%0 Journal Article %A V. V. Kozlov %A D. V. Treschev %T Topology of the configuration space, singularities of the~potential, and polynomial integrals of equations of dynamics %J Sbornik. Mathematics %D 2016 %P 1435-1449 %V 207 %N 10 %I mathdoc %U http://geodesic.mathdoc.fr/item/SM_2016_207_10_a4/ %G en %F SM_2016_207_10_a4
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/