分数阶约束力学系统理论

The differences and relations of the several definitions of fractional derivative and integral are discussed.The classification and unified for kinds of definitions of fractional derivative and integral are researched. Analyzing the characteristic of four types fractional variations principles.A new type variation principles of fractional constrained mechanical systems is presented. The ration model describing constrained mechanical systems are researched. The theory of Noether and Lie symmetries is studied, a expression of conserved quantity is deduced,which is constant on the trajectory of system. Investigation of the project further strengthen application foundations of fractional calculus, expand its application area ,improve and perfect the theory of fractional constrained mechanical systems. The research of these items not only has important theoretical values, but also has deep practical background.

研究几种常用的分数阶微积分定义的区别与联系，探索各种分数阶微积分定义的分类与统一；分析四类分数阶变分原理的特点，提出一种新型分数阶约束力学系统的变分原理，探索利用分数阶微积分工具描述约束力学系统的合理模型；进一步研究分数阶约束力学系统的Noether对称性和Lie对称性理论，提出在系统的运动轨迹上要保持为常量的分数阶守恒量表达式。课题研究将进一步筑牢分数阶微积分的应用基础，拓展分数阶微积分在动力学系统中的应用范围，完善分数阶约束力学系统理论。因此，不仅具有重要的理论价值，也具有重要的应用价值。

项目是将分数阶微积分工具应用于约束力学系统，主要研究了：分数阶算子的积分性质；广义变分原理和Birkhoff方程；分数阶约束力学系统的对称性理论； Herglotz变分原理在分数阶约束系统中的应用问题等。给出了Riemann-Liouville, Caputo,Riesz等分数阶导数的积分性质；并应用新算子构造了广义Pfaff变分原理，并导出包含新算子的分数阶Birkhoff方程；分别研究了Lagrange系统、 Hamilton 系统和Birkhoff系统的分数阶Noether理论；提出了Herglotz型分数阶基于Birkhoff系统的变分原理和Noether定理。本项目一个重要工作是导出了关于广义分数阶导数算子的两个“变换公式”。其意义在于利用这两个“变换公式”可以得到显式形式的分数阶守恒量，是分数阶对称理论研究中的一个突破。