The dynamical behavior and physical process of a complex system can be described and studied more ac-curately by using a fractional model, at the same time the Birkhoffian mechanics is a generalization of Hamiltonian mechanics, and therefore, the study of dynamics of fractional Birkhoffian systems is of great significance. Fractional Noether's theorem reveals the intrinsic relation between the Noether symmetric transformation and the fractional con-served quantity, but when the transformation is replaced by the Noether quasi-symmetric transformation, the correspond-ing extension of Noether's theorem is very difficult. In this paper, the Noether quasi-symmetry and the conserved quantity for fractional Birkhoffian systems in terms of Caputo derivatives are presented and studied by using a technique of time-reparametrization. Firstly, the technique is applied to the study of the Noether quasi-symmetry and the conserved quantity for classical Birkhoffian systems and Noether's theorem in its general form is established. Secondly, the definitions and criteria of Noether quasi-symmetric transformations for fractional Birkhoffian systems are given which are based on the invariance of fractional Pfaff action under one-parameter infinitesimal group of transformations without transforming the time and with transforming the time, respectively. Based on the definition of fractional conserved quantity proposed by Frederico and Torres, Noether's theorem for fractional Birkhoffian systems is established by using the method of time-reparametrization. The theorem reveals the inner relationship between Noether quasi-symmetry and fractional conserved quantity and contains Noether's theorem for the symmetry of fractional Birkhoffian system and Noether's theorem for classical Birkhoffian system as its specials. Finally, we take the Hojman-Urrutia problem as an example to illustrate the application of the results.%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Hamil-ton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.
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