分数阶非保守Lagrange系统的一类新型绝热不变量  

A new type of adiabatic invariant for fractional order non-conservative Lagrangian systems

作　　者：徐鑫鑫 张毅[2] Xu Xin-Xin;Zhang Yi(School of Mathematics and Physics,Suzhou University of Science and Technology,Suzhou 215009,China;School of Civil Engineering,Suzhou University of Science and Technology,Suzhou 215011,China)

机构地区：[1]苏州科技大学数理学院,苏州215009 [2]苏州科技大学土木工程学院,苏州215011

出　　处：《物理学报》2020年第22期285-292,共8页Acta Physica Sinica

基　　金：国家自然科学基金(批准号:11972241,11572212,11272227);江苏省自然科学基金(批准号:BK20191454);江苏省普通高校研究生科研创新计划(批准号:KYCX18_2548)资助的课题。

摘　　要：为了更加准确地描述复杂非保守系统的动力学行为,将Herglotz变分原理推广到分数阶模型,研究分数阶非保守Lagrange系统的绝热不变量.首先,基于Herglotz变分问题,导出分数阶非保守Lagrange系统的Herglotz型微分变分原理并进一步得到分数阶非保守Lagrange系统的运动微分方程;其次,引进无限小单参数变换,由等时变分和非等时变分的关系,导出了分数阶非保守Lagrange系统的Herglotz型精确不变量;再次,研究小扰动对分数阶Lagrange系统的影响,建立了基于Caputo导数的分数阶Lagrange系统的绝热不变量存在的条件,得到了该系统的Herglotz型绝热不变量;最后,举例说明结果的应用.!1 The Herglotz variational problem is also known as Herglotz generalized variational principle whose action functional is defined by differential equation.Unlike the classical variational principle,the Herglotz variational principle gives a variational description of a holonomic non-conservative system.The Herglotz variational principle can describe not only all physical processes that can be described by the classical variational principlen,but also the problems that the classical variational principle is not applicable for.If the Lagrangian or Hamiltonian does not depend on the action functional,the Herglotz variational principle reduces to the classical integral variational principle.In this work,in order to describe the dynamical behavior of complex nonconservative system more accurately,we extend the Herglotz variational principle to the fractional order model,and study the adiabatic invariant for fractional order non-conservative Lagrangian system.Firstly,based on the Herglotz variational problem,the differential variational principle of Herglotz type and the differential equations of motion of the fractional non-conservative Lagrangian system are derived.Secondly,according to the relationship between the isochronal variation and the nonisochronal variation,the transformation of invariance condition of Herglotz differential variational principle is established and the exact invariants of the system are derived.Thirdly,the effects of small perturbations on fractional non-conservative Lagrangian systems are studied,the conditions for the existence of adiabatic invariants for the Lagrangian systems of Herglotz type based on Caputo derivatives are established,and the adiabatic invariants of Herglotz type are obtained.In addition,the exact invariant and adiabatic invariant of fractional non-conservative Hamiltonian system can be obtained by Legendre transformation.Whenα→1,the Herglotz differential variational principle for fractional non-conservative Lagrangian system degrades into classical Herglotz differe

关 键 词：非保守Lagrange系统 Herglotz广义变分原理 不变量 分数阶微积分 

分 类 号：O316[理学—一般力学与力学基础]