一类转动系统中质点的不变环面运动存在性问题  

Existence Problem of Invariant Torus Particle Motion in Rotating Nonlinear Dynamical Systems

作　　者：王璟 谢建华[1] 乐源[1] 

机构地区：[1]西南交通大学力学与工程学院,四川成都610031

出　　处：《西南交通大学学报》2017年第5期1015-1019,共5页Journal of Southwest Jiaotong University

基　　金：国家自然科学基金资助项目(11272268;11172246)

摘　　要：为研究可积哈密顿系统的不变环面在小扰动下的保持性问题,建立了极坐标系下圆盘转动系统的哈密顿方程.首先,通过能量守恒的初积分将两自由度系统转化为二阶状态变量方程形式的单自由度系统;其次,在此基础上,利用KAM(Kolmogorov-Arnold-Moser)定理证明了不变环面的存在性;最后,对圆盘转动系统的动力学特性进行了数值模拟,结果表明:系统的时程曲线是周期的,相图稠密环绕,庞加莱映射为一条闭曲线;系统做拟周期运动,可积哈密顿系统的不变环面在小扰动下仍然存在,庞加莱映射的闭曲线对应着系统的KAM不变环面.In order to study whether the invariant torus of integrable Hamiltonian systems is retained under small perturbations, we established the Hamiltonian equations in polar coordinates. Using the first integral of the energy conservation equation, the transformation of the second-order state variable from a system with two degrees of freedom into a system with a single degree of freedom was analysed. Secondly, based on the Kolmogorov -Arnold-Moser （KAM） theorem, the existence of invariant tori in the perturbed system was confirmed. Finally, numerical simulations were performed to elucidate the analysis. The results show that the time history curve of the system is periodic, the phase portrait is dense, and the Poincare map is a closed curve. The system is quasi- periodic, and the invariant torus of the integrable Hamiltonian system is shown to still exist under small perturbations. Moreover, the closed curve Poincare mapping corresponds to the KAM invariant closed curve.

关 键 词：哈密顿系统 KAM理论 不变环面 

分 类 号：O313[理学—一般力学与力学基础]