Generalized Canonical Transformations for Fractional Birkhoffian Systems

分数阶Birkhoff系统的广义正则变换

This paper presents fractional generalized canonical transformations for fractional Birkhoffian systems within Caputo derivatives.Firstly,based on fractional Pfaff-Birkhoff principle within Caputo derivatives,fractional Birkhoff’s equations are derived and the basic identity of constructing generalized canonical transformations is proposed.Secondly,according to the fact that the generating functions contain new and old variables,four kinds of generating functions of the fractional Birkhoffian system are proposed,and four basic forms of fractional generalized canonical transformations are deduced.Then,fractional canonical transformations for fractional Hamiltonian system are given.Some interesting examples are finally listed.

研究给出Caputo导数下分数阶Birkhoff系统的分数阶广义正则变换。首先,基于Cuputo导数下分数阶Pfaff-Birkhoff原理,导出分数阶Birkhoff方程,建立构造广义正则变换的基本恒等式。其次,根据母函数含有新、旧变量的情况,提出分数阶Birkhoff系统的4类母函数,并导出相应的4种基本形式的分数阶广义正则变换。再次,给出分数阶Hamilton系统的分数阶正则变换。最后给出若干有趣的算例。