具有非标准Lagrange函数的分数阶动力学系统的正则变换  

Canonical transformations for fractional dynamical systems with non-standard Lagrangians

作　　者：张毅[1] 朱琳 ZHANG Yi;ZHU Lin(School of Civil Engineering,SUST,Suzhou 215011,China;School of Mathematical Sciences,SUST,Suzhou 215009,China)

机构地区：[1]苏州科技大学土木工程学院,江苏苏州215011 [2]苏州科技大学数学科学学院,江苏苏州215009

出　　处：《苏州科技大学学报（自然科学版）》2024年第4期60-68,共9页Journal of Suzhou University of Science and Technology（Natural Science Edition）

基　　金：国家自然科学基金项目(12272248)。

摘　　要：分数阶模型由于所具有的系统开放性、时间依赖性和空间全域性,为复杂物理、力学和工程问题的建模提供了有力的数学工具。作为分析力学的重要内容之一,变换在方程简化与求解过程中发挥重要作用,通过正则变换可使原本难以求解的方程得到简化且变换后仍保持原方程的形式。因此,研究分数阶动力学系统的正则变换具有重要意义。文章研究具有非标准Lagrange函数(包括指数Lagrange函数、幂律Lagrange函数和对数Lagrange函数)的分数阶动力学系统的正则变换。首先,定义分数阶广义动量和非标准Hamilton函数,导出三类非标准分数阶动力学系统的Hamilton正则方程;其次,建立非标准分数阶动力学系统的正则变换的判据方程,依据母函数的不同选择给出四种基本形式的正则变换;最后,通过若干算例介绍结果的应用。The fractional-order model provides a powerful mathematical tool for the modeling of complex physics,mechanics and engineering problems because of its system openness,time dependence and spatial localization.As one of the important contents of analytical mechanics,transformation plays an irreplaceable role in the process of equation simplification and solution.Through canonical transformations,the equations difficult to be solved can be simplified and the form of the original equations remains after transformations.Therefore,it is of great significance to study the canonical transformation of fractional order dynamical systems.This paper studies the canonical transformations of fractional dynamical systems with non-standard Lagrangians,including exponential,power-law and logarithmic Lagrangians.Firstly,fractional generalized momentums and non-standard Hamiltonians were defined,and Hamilton canonical equations of three kinds of non-standard fractional dynamical systems were derived.Secondly,the criterial equations of canonical transformations of non-standard fractional dynamical systems were established,and four basic forms of canonical transformations were given according to different choices of generating functions.Finally,some examples were given to illustrate the application of the results.

关 键 词：非标准Lagrange函数 分数阶动力学系统 正则方程 正则变换 

分 类 号：O316[理学—一般力学与力学基础]