基于分数阶模型的约束力学系统变分问题与对称性研究

The present research.①studies the fractional variational problems of constrained systems to establish corresponding fractional Lagrange equations, fractional Hamilton canonical equations and their transversality conditions. It analyzes the effects of different fractional order models, different fractional derivatives, and different boundary conditions or initial conditions on results. The fractional variational problems for the Lagrangian containing delay terms, the fractional variational problems with variable order fractional operators, and the multitime fractional variational problems are focal points of this research work. .②studies the fractional integration theory of constrained systems based on different fractional models, different fractional derivatives, and different boundary conditions or initial conditions, including fractional Hamilton-Jacobi method, the theory of fractional canonical transformations, fractional invariants of integration and the Poisson theory with fractional derivatives. .③studies the fractional variational symmetry of constrained systems, analyzes the effects of constraints on the variational symmetry, defines the fractional conserved quantities appropriately, and establishes the Noether theory of constrained systems based on different fractional models and different fractional derivatives. The fractional variational symmetry and fractional conserved quantities for the Lagrangian containing delay terms, with variable order fractional operators or multitime functional are also focal points of this research work. .This research is conducted in an effort to improve and perfect the theoretical system of fractional variational problems, to provide a new approach to researching and solving the complex dynamic problems of constrained systems, to open up a new field for applying fractional calculus theory and ultimately to promote the researches on dynamics of constrained systems to a higher level.

研究内容：①研究约束系统的分数阶变分问题，建立相应的分数阶拉格朗日方程、分数阶哈密顿正则方程及其相截性条件．分析不同分数阶模型、不同分数阶导数定义、不同的边界条件或初始条件对结果的影响．重点研究含有延迟项、可变阶或多重泛函的分数阶变分问题．②研究约束系统基于不同分数阶模型、不同分数阶导数定义和不同的边界条件或初始条件的分数阶积分理论，包括分数阶哈密顿-雅可比方法、分数阶正则变换理论、分数阶积分不变量和分数阶泊松理论．③研究约束系统的分数阶变分对称性，分析约束对变分对称性的影响，恰当定义分数阶守恒量，建立约束系统基于不同分数阶模型、不同分数阶导数定义的诺特理论．重点研究含有延迟项、可变阶或多重泛函的分数阶变分对称性与守恒量．研究意义：①充实和完善分数阶变分问题的理论体系；②提供研究和解决约束系统复杂动力学问题的新途径；③开辟分数阶微积分理论应用的新领域；④促使约束系统动力学研究上一个新台阶．

分数阶模型是建立在分数阶微积分和分数阶微分方程基础上的数学模型。因为可以简洁、准确地描述具有历史记忆性和空间非局域相关性等力学与物理过程，且分数阶导数建模简单、参数物理意义清楚、描述准确，因而分数阶微积分成为复杂力学与物理过程数学建模的重要工具之一，已被广泛应用于解决科学和工程诸多领域的问题。本项目提出并研究了完整系统、非完整系统和伯克霍夫系统基于阿加瓦尔模型、阿塔纳兹库维奇模型和埃尔-纳卜茜模型的分数阶变分问题；提出并深入研究了含时滞的完整和非完整系统、伯克霍夫系统的分数阶变分问题、变导数分数阶伐夫-伯克霍夫变分问题；研究了在卡普托导数和联合卡普托导数下分数阶力学系统的变换理论，给出了四种基本形式的分数阶正则变换；定义了分数阶拉格朗日系统的积分因子，并基于积分因子构造了系统的守恒律；分别基于弗雷德里克-托里斯分数阶守恒量定义和经典意义的守恒量概念，提出并建立了黎曼-刘维尔导数、卡普托导数和黎兹导数下分数阶伯克霍夫系统的诺特对称性理论；深入研究了埃尔-纳卜茜分数阶模型下完整和非完整系统、伯克霍夫系统的诺特对称性与守恒量；提出并建立了分数阶模型下含时滞的完整和非完整系统、伯克霍夫系统的诺特定理；研究了变导数分数阶伯克霍夫系统的诺特对称性与守恒量；研究了阿塔纳兹库维奇模型和埃尔-纳卜茜模型下分数阶伯克霍夫系统和完整系统对称性的摄动与绝热不变量；研究了分数阶力学系统的李对称性和梅对称性；提出并研究了基于非标准拉格朗日函数的动力学系统的对称性与守恒量；提出并建立了伯克霍夫系统和非保守哈密顿系统的海格罗兹广义变分原理与诺特定理及其逆定理；提出并研究了时间尺度上伯克霍夫系统和哈密顿系统的诺特对称性与守恒量。本项目的研究为解决约束力学系统复杂动力学问题提供了新思路和新途径，同时开辟了分数阶微积分理论应用的新领域。这些研究都属本研究方向内的创新点，所取得的研究成果整体上与国际先进水平同步。