分数阶动力学的分析力学方法及其应用

综述我们在分数阶动力学分析力学方法的研究进展,包括：分数阶动力学系统的分析力学表示,构造分数阶动力学模型的分析力学方法,构造分数阶动力学模型团簇的分析力学方法,三类分数阶Lie群无限小变换方法,分数阶动力学系统的对称性、对称性摄动和共形不变性的分析力学方法,分数阶动力学系统的代数结构与Poisson积分的分析力学方法,构造分数阶动力学系统积分不变量的分析力学方法,分数阶动力学系统梯度表示的分析力学方法,分数阶动力学系统稳定性的分析力学方法,分数阶微分方程的分析力学方法等,介绍了对于物理学、力学、生物学、非线性科学等领域的10多种分数阶动力学模型的应用,并指出了若干进一步研究的问题.

Analytical mechanics method of fractional dynamics and its applications

A concise review on analytical mechanics methods for fractional dynamics was presented. It mainly focused on the analytical mechanics representations of fractional dynamical systems, the analytical mechanics methods for constructing a fractional dynamical model and a family of fractional dynamical models, three types of fractional infinitesimal transformation for Lie group, the analytical mechanics methods for symmetry, symmetrical perturbation, conformal invariance, algebraic structure, Poisson integral, integral invariant, gradient representation, stability of fractional dynamical systems, and the analytical mechanics method of fractional differential equations. The applications of more than 10 kinds of fractional dynamics models in physics, mechanics, biology and non linear science were introduced. It also pointed out some problems should be studied further in the future.