Fractional Birkhoffian mechanics
Fractional Birkhoffian mechanics
复制标题
DOI：
10.1007/s00707-014-1230-1
复制
发表时间：
2015-03
期刊：
影响因子：
2.7
通讯作者：
S. Luo;Yan-Li Xu
中科院分区：
文献类型：
--
作者：
S. Luo;Yan-Li Xu
本文提出了一种新的分数阶动力学理论，即具有分数阶导数的Birkhoff系统的动力学(分数阶Birkhoff力学)，它给出了构造实际问题的分数阶动力学模型的一般方法。利用组合分数导数的定义，给出了一个统一的分数Pfaff作用量和一个统一的分数Pfaff-Birkhoff原理，并在不同的分数导数定义下给出了四种分数Pfaff-Birkhoff原理。然后，利用分数阶Pfaff-Birkhoff原理，我们建立了一系列具有不同分数导数的分数阶Birkhoff方程，并构造了自治分数阶Birkhoff方程的张量表示。进一步，我们研究了分数阶Bikhoff系统、分数阶哈密顿系统和分数阶Lagrangian系统之间的关系，并给出了变换条件。此外，作为分数阶Birkhoff方法的应用，我们构造了五种分数阶动力学模型，包括分数阶Lotka生化振子模型、分数阶Whittaker模型、分数阶Hojman-Urrutia模型、分数阶Hénon-Heiles模型和分数阶Emden模型。
In this paper, we present a new fractional dynamical theory, i.e., the dynamics of a Birkhoffian system with fractional derivatives (the fractional Birkhoffian mechanics), which gives a general method for constructing a fractional dynamical model of the actual problem. By using the definition of combined fractional derivative, we present a unified fractional Pfaff action and a unified fractional Pfaff–Birkhoff principle, and give four kinds of fractional Pfaff–Birkhoff principles under the different definitions of fractional derivative. And then, by using the fractional Pfaff–Birkhoff principles, we establish a series of fractional Birkhoffian equations with different fractional derivatives and construct the tensor representation of autonomous fractional Birkhoffian equations. Further, we study the relationship among the fractional Bikhoffian system, the fractional Hamiltonian system and the fractional Lagrangian system and give the transformation conditions. And furthermore, as applications of the fractional Birkhoffian method, we construct five kinds of fractional dynamical models, which include the fractional Lotka biochemical oscillator model, the fractional Whittaker model, the fractional Hojman–Urrutia model, the fractional Hénon–Heiles model and the fractional Emden model.