Lie Symmetries and Conserved Quantities of the Fractional Singular Systems
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摘要:

对称性与守恒量可以简化动力学问题从而进一步求出力学系统的精确解,这样更加有利于研究动力学行为.分数阶模型相比于整数阶模型,能够描述复杂系统的动力学过程,因此在分数阶模型下研究对称性与守恒量是不可或缺的.首先介绍两个分数阶奇异系统,一个系统包含混合整数和Caputo分数阶导数,另一个系统仅含Caputo分数阶导数.由两个分数阶奇异系统分别给出两个分数阶固有约束,并给出对应的分数阶约束Hamilton方程.然后,基于微分方程在无限小变换下的不变性,给出了分数阶约束Hamilton方程Lie对称性的定义,导出了相应的确定方程,限制方程和附加限制方程.第三,建立并证明了两个分数阶约束Hamilton系统的Lie对称性定理,得到了相应的分数阶约束Hamilton系统的Lie守恒量.在特定条件下,本文所得结果可以退化为整数阶约束Hamilton系统的Lie守恒量.最后通过两个算例来说明此结果的应用.

Abstract:

Symmetry and conserved quantity can simplify the dynamic problem and further obtain the exact solution of the mechanical system, which is more conducive to the study of dynamic behavior. Compared with the integer order model, the fractional model can describe the dynamic process of complex systems. Therefore, it is indispensable to study the symmetry and conserved quantities under the fractional model. Firstly, two fractional singular systems are introduced. One system contains mixed integers and Caputo fractional derivatives, and the other system contains only Caputo fractional derivatives. Two fractional inherent constraints are given by two fractional singular systems, and the corresponding fractional constrained Hamilton equation is given. Then, based on the invariance of differential equation under infinitesimal transformation, the definition of Lie symmetry of fractional constrained Hamilton equation is given, and the corresponding determined equation, restriction equation and additional constraint equation are derived. Thirdly, the Lie symmetry theorems of two fractional constrained Hamiltonian systems are established and proved, and the Lie conserved quantities of the corresponding fractional constrained Hamiltonian systems are obtained. Under certain conditions, the results obtained in this paper can be reduced to Lie conserved quantities of integer order constrained Hamiltonian systems. Finally, two examples are given to illustrate the application of this result.

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##### 历史
• 收稿日期:2022-01-25
• 最后修改日期:2022-04-21
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• 在线发布日期: 2023-10-23
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