具有高次多项式积分的动力学方程的离散对称性

IF 0.8 3区 数学 Q2 MATHEMATICS
Valery Vasil'evich Kozlov
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引用次数: 1

摘要

我们考虑具有环面构型空间和动能的系统,其形式为环面上的“平坦”黎曼度规。势能V是位形环面上的光滑函数。这种系统的动力学用微分方程的“自然”哈密顿系统来描述。如果$V$被$\varepsilon V$代替,其中$\varepsilon$是一个小参数,那么研究这种小$\varepsilon$的哈密顿系统是根据poincar的“动力学主要问题”的一部分。讨论了关于运动方程单值动量-多项式积分存在性的著名猜想:如果存在一个动量-多项式积分,则存在一个动量-线性积分或动量-二次积分。对于$m=3$和$m=4$,证明了这个猜想的完全一般性。我们研究了“较高”度$m=5$和$m=6$的情况。与哈密顿系统的微扰理论类似,我们在动量平面上引入了共振线。如果一个系统允许多项式积分,那么这些线的数目是有限的。发现了共振线集合的对称性,并由此导出了可积性的必要条件。给出了单值多项式积分存在性的几个新判据。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Discrete symmetries of equations of dynamics with polynomial integrals of higher degrees
We consider systems with toric configuration space and kinetic energy in the form of a "flat" Riemannian metric on the torus. The potential energy $V$ is a smooth function on the configuration torus. The dynamics of such systems is described by "natural" Hamiltonian systems of differential equations. If $V$ is replaced by $\varepsilon V$, where $\varepsilon$ is a small parameter, then the study of such Hamiltonian systems for small $\varepsilon$ is a part of the "main problem of dynamics" according to Poincaré. We discuss the well-known conjecture on the existence of single-valued momentum-polynomial integrals of motion equations: if there is a momentum-polynomial integral of degree $m$, then there exist a momentum-linear or momentum-quadratic integral. This conjecture was verified in full generality for $m=3$ and $m=4$. We study the cases of "higher" degrees $m=5$ and $m=6$. Similarly to the theory of perturbations of Hamiltonian systems, we introduce resonance lines on the momentum plane. If a system admits a polynomial integral, then the number of these lines is finite. The symmetries of the set of resonance lines are found, from which, in particular, necessary conditions for integrability are derived. Some new criteria for the existence of single-valued polynomial integrals are obtained.
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来源期刊
Izvestiya Mathematics
Izvestiya Mathematics 数学-数学
CiteScore
1.30
自引率
0.00%
发文量
30
审稿时长
6-12 weeks
期刊介绍: The Russian original is rigorously refereed in Russia and the translations are carefully scrutinised and edited by the London Mathematical Society. This publication covers all fields of mathematics, but special attention is given to: Algebra; Mathematical logic; Number theory; Mathematical analysis; Geometry; Topology; Differential equations.
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