时间尺度上广义Birkhoff系统Lie对称性摄动  

PERTURBATION TO LIE SYMMETRIES FOR GENERALIZED BIRKHOFFIAN SYSTEMS ON TIME SCALES

作　　者：侯爽 宋传静[1] Hou Shuang;Song Chuanjing(School of Mathematical Sciences,Suzhou University of Science and Technology,Suzhou 215009,Jiangsu,China)

机构地区：[1]苏州科技大学数学科学学院,江苏苏州215009

出　　处：《力学学报》2024年第12期3591-3600,共10页Chinese Journal of Theoretical and Applied Mechanics

基　　金：国家自然科学基金(12172241,12272248);江苏高校“青蓝工程”资助项目。

摘　　要：近几年,在时间尺度框架下研究对称性与守恒量日益引发关注.然而,因该领域尚处于探索阶段,其研究成果的可靠性仍在进一步考察中,且对称性的摄动与绝热不变量的研究也正是基于此项工作之上,因此对于该领域的深入探讨具有重要意义.首先,在时间尺度框架下,给出了精确不变量与绝热不变量的定义,对于未受扰动力作用和受扰动力作用的广义Birkhoff系统,分别建立了Lie对称性及其摄动的确定方程和结构方程,并基于此得到了该系统Lie对称性导致的精确不变量及其摄动导致的绝热不变量,并给出相应证明.其次,考虑受约束Birkhoff系统,对于未受扰动和受扰动的时间尺度上约束Birkhoff系统及其相应自由Birkhoff系统,分别给出了Lie对称性导致精确不变量及其摄动导致绝热不变量的条件.相应小节末尾分别给出算例并对所得守恒量进行了数值模拟,直观地验证了结论的有效性.取时间尺度为实数集和整数集,所有结论可退化到经典连续型和离散型动力学系统.本论文的方法与成果对时间尺度动力学系统对称性及其摄动理论研究具有一定的指导意义,可应用和拓展到非迁移系统,对偶系统,分数阶时间尺度相结合系统,nabla导数情形等.In recent years,the study of symmetry and conserved quantity under time-scale framework has increasingly attracted attention.Nevertheless,as this field is still in the exploratory stage,the reliability of its research results is still under further investigation,and the study of symmetry perturbations and adiabatic invariants is also based on this work.Therefore,in-depth exploration of this field is of great significance.First of all,under time-scale framework,the definitions of exact invariants and adiabatic invariants are given.For both undisturbed and disturbed generalized Birkhoffian systems,the determining equations and structural equations for Lie symmetry and its perturbations are established,respectively.Based on these,the exact invariants caused by Lie symmetry and the adiabatic invariants caused by perturbations to Lie symmetry in the both systems are obtained,and the corresponding proofs are given.Secondly,regarding the Birkhoffian systems under constrained conditions on time scales,we explore the conditions under which the Lie symmetry of the constrained Birkhoffian system and its corresponding free Birkhoffian system leads to exact invariants,as well as the conditions under which the perturbation to Lie symmetry of the corresponding disturbed systems leads to adiabatic invariants.At the end of the corresponding sections,examples are provided and numerical simulations are conducted on the obtained conserved quantities,which intuitively verified the effectiveness of the conclusions in this paper.Taking the time scale as the set of real numbers and integer,all conclusions in this paper can be degraded to classical continuous and discrete dynamical systems.The methods and research results of this paper have certain reference and guiding significance for the study of symmetry and perturbation theory of dynamical systems on time scales,and can be applied and extended to nonshifted systems,dual systems,the combination of fractional and time-scale systems,and nabla derivative cases,and so on.

关 键 词：时间尺度 LIE对称性 摄动与绝热不变量 BIRKHOFF系统 

分 类 号：O316[理学—一般力学与力学基础]