Caputo导数下分数阶Birkhoff系统的准对称性与分数阶Noether定理

发布时间：2018-12-13 17:53

【摘要】：应用分数阶模型可以更准确地描述和研究复杂系统的动力学行为和物理过程,同时Birkhoff力学是Hamilton力学的推广,因此研究分数阶Birkhoff系统动力学具有重要意义.分数阶Noether定理揭示了Noether对称变换与分数阶守恒量之间的内在联系,但是当变换拓展为Noether准对称变换时,该定理的推广遇到了很大的困难.本文基于时间重新参数化方法提出并研究Caputo导数下分数阶Birkhoff系统的Noether准对称性与守恒量.首先,将时间重新参数化方法应用于经典Birkhoff系统的Noether准对称性与守恒量研究,建立了相应的Noether定理;其次,基于分数阶Pfaff作用量分别在时间不变的和一般单参数无限小变换群下的不变性给出分数阶Birkhoff系统的Noether准对称变换的定义和判据,基于Frederico和Torres提出的分数阶守恒量定义,利用时间重新参数化方法建立了分数阶Birkhoff系统的Noether定理,从而揭示了分数阶Birkhoff系统的Noether准对称性与分数阶守恒量之间的内在联系.分数阶Birkhoff系统的Noether对称性定理和经典Birkhoff系统的Noether定理是其特例.最后以分数阶Hojman-Urrutia问题为例说明结果的应用.
[Abstract]:The fractional order model can be used to describe and study the dynamics and physical processes of complex systems more accurately, and Birkhoff mechanics is a generalization of Hamilton mechanics, so it is of great significance to study the dynamics of fractional Birkhoff systems. The fractional order Noether theorem reveals the intrinsic relationship between the Noether symmetric transformation and fractional order conserved quantities, but when the transformation is extended to the Noether quasi-symmetric transformation, the generalization of the theorem meets great difficulties. In this paper, we present and study the Noether quasi-symmetry and conserved quantities of fractional Birkhoff systems under Caputo derivatives based on the time re-parameterization method. Firstly, the time reparameterization method is applied to the study of Noether quasi-symmetry and conserved quantity of classical Birkhoff system, and the corresponding Noether theorem is established. Secondly, based on the invariance of fractional order Pfaff action under time invariant and general single parameter infinitesimal transformation group, the definition and criterion of Noether quasi-symmetric transformation for fractional Birkhoff system are given, and the fractional order conserved quantity is defined by Frederico and Torres. The Noether theorem of fractional Birkhoff system is established by using time reparameterization method, and the intrinsic relation between Noether quasi symmetry and fractional order conserved quantity of fractional Birkhoff system is revealed. The Noether symmetry theorem of fractional Birkhoff system and the Noether theorem of classical Birkhoff system are special cases. Finally, the application of the result is illustrated by an example of fractional order Hojman-Urrutia problem.
【作者单位】：苏州科技大学土木工程学院;
【基金】：国家自然科学基金资助项目(11272227,11572212)
【分类号】：O316

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