TY - JOUR

T1 - On the Rational Integrals of Two-Dimensional Natural Systems

AU - Agapov, S. V.

AU - Tursunov, M. M.

N1 - The first author was supported by the Russian Science Foundation (Grant no. 19–11–00044-P).

PY - 2023/7

Y1 - 2023/7

N2 - We study a natural mechanical system having an additional first integral in the formof a function rational in momenta. One of the authors has proved recently that ifthe configuration space of the system is the two-dimensional torus; then, provided thatthe potential is analytic, the existence of a rational integral withanalytic periodic coefficients and small degrees of the numerator and denominator impliesthe existence of an integral linear in momenta. In the present article, this resultis generalized to the case that the configuration space of the system isthe two-dimensional plane.

AB - We study a natural mechanical system having an additional first integral in the formof a function rational in momenta. One of the authors has proved recently that ifthe configuration space of the system is the two-dimensional torus; then, provided thatthe potential is analytic, the existence of a rational integral withanalytic periodic coefficients and small degrees of the numerator and denominator impliesthe existence of an integral linear in momenta. In the present article, this resultis generalized to the case that the configuration space of the system isthe two-dimensional plane.

KW - 517.938

KW - Hopf equation

KW - first integral rational in momenta

KW - natural system

KW - potential

UR - https://www.scopus.com/record/display.uri?eid=2-s2.0-85165587047&origin=inward&txGid=e0172f1564fad384f73cf5788825d671

UR - https://www.mendeley.com/catalogue/562c127b-6ae4-3003-95af-e332a4679f8c/

U2 - 10.1134/S0037446623040018

DO - 10.1134/S0037446623040018

M3 - Article

VL - 64

SP - 787

EP - 795

JO - Siberian Mathematical Journal

JF - Siberian Mathematical Journal

SN - 0037-4466

IS - 4

ER -