分数阶力学系统的正则变换理论  

Theiry of Canonical Transformation for a Fractional Mechanical System

作　　者：张毅[1] 

机构地区：[1]苏州科技大学土木工程学院,苏州215011

出　　处：《应用数学学报》2016年第2期249-260,共12页Acta Mathematicae Applicatae Sinica

基　　金：国家自然科学基金(No.10972151;11272227;11572212)资助项目

摘　　要：应用分数阶模型可以更准确地描述复杂系统的力学与物理行为,随着分数阶微积分在科学和工程的诸多领域的成功应用,传统的分析力学理论和方法需要拓展到含有分数阶微积分的系统.变换是分析力学研究的一个重要手段.本文研究分数阶力学系统的变换理论.基于Cuputo分数阶导数的定义,定义力学系统的Lagrange函数和Hamilton函数,在H(o|¨)lder交换关系下建立了分数阶Hamilton原理,并由分数阶Hamilton原理通过变分运算导出分数阶Hamilton正则方程;建立了分数阶力学系统的正则变换理论,给出了四种基本形式的分数阶正则变换,并通过算例说明母函数在分数阶正则变换中的作用.The application of fractional model can be more accurately to describe the mechanical and physical behavior of a complex system. With the fractional calculus used successfully in many areas of science and engineering, the traditional theories and methods of analytical mechanics need to be extended to the systems with fractional calculus. Trans- formation is a vital tool in the study of analytical mechanics. This paper focuses on studying the theory of transformation for a fractional mechanical system. Based upon the definition of Caputo fractional derivative, the Lagrangian and the Hamiltonian of a mechanical system is defined, and the fractional Hamilton principle is established under the exchange rela- tionship of HSlder, and the fractional Hamilton canonical equations are deduced by means of variational calculation from the fractional Hamilton principle. The theory of canonical transformation for the fractional systems is established, and four basic forms of fractional canonical transformation are given, and some examples are given to illustrate the application of the results and the role played by a generating function in the canonical transformation.

关 键 词：分数阶力学系统 分数阶变分问题 正则方程 正则变换 

分 类 号：O175.1[理学—数学] O316[理学—基础数学]