分数阶Birkhoff系统的Lie对称性与守恒量

Lie symmetry and conserved quantity for fractional Birkhoffian system

为了进一步揭示力学系统的对称性与守恒量之间的内在关系,提出并研究了基于Riemann-Liouville导数的分数阶Birkhoff系统的Lie对称性与守恒量。首先,建立了分数阶Birkhoff方程,基于微分方程在无限小变换下的不变性,给出了分数阶Birkhoff系统Lie对称性的确定方程;其次,给出了分数阶Birkhoff系统的Lie对称性与守恒量定理。定理表明:当无限小变换的生成元满足结构方程时,由Lie对称性可找到系统的守恒量。经典Birkhoff系统和分数阶Hamilton系统的Lie对称性与守恒量定理是该文之特例。文末,给出两个算例以说明结果的应用。

In order to further reveal the inner relationship between the symmetry and the conserved quantity of a dynamical system, this paper investigates the Lie symmetry and the conserved quantity for the fractional Birkhoffian system in terms of Riemann-Liouville derivatives. Firstly,the fractional Birkhoff＇s equations were estab-lished. Based on the invariance of the differential equations under the infinitesimal transformation, the determin- ing equations of Lie symmetry for the fractional Birkhoffian system were given. Secondly, the theorem of Lie sym- metry and conserved quantity for the fractional Birkhoffian system was presented. The theorem shows if the generators of the infinitesimal transformation satisfy the structure equations, the conserved quantity can be found by the Lie symmetry of the system. The results contain the theorems of Lie symmetries and conserved quantities for the classical Birkhoffian systems and the fractional Hamiltonian systems as specials. At the end of the paper, two examples were given to illustrate the application of the results.