Abstract

The problem of first integrals that are polynomial in momenta is considered for the equations of motion of a particle on a two-dimensional Euclidean torus in a force field with even potential. Of special interest is the case when the spectrum of the potential lies on four straight lines such that the angle between any two of them is a multiple of \(\pi/4\). With the help of perturbation theory, it is proved that there are no additional polynomial integrals of any degree that are independent of the Hamiltonian function.

中文翻译：

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一类可逆哈密顿系统的动量-多项式积分

摘要

对于具有偶数势的力场中的二维欧几里得环面上的粒子运动方程，考虑了矩量为多项式的第一积分的问题。当电势的频谱位于四条直线上，使得它们之间的任意两个角度为\（\ pi / 4 \）的倍数时，会引起特别的关注。借助摄动理论，证明没有独立于哈密顿函数的任何程度的附加多项式积分。