Existence Problem of Invariant Torus Particle Motion in Rotating Nonlinear Dynamical Systems
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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.Abstract: 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 Poincaré 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 Poincaré mapping corresponds to the KAM invariant closed curve.
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Key words:
- Hamiltonian system /
- KAM theory /
- invariant torus
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THOMAS K. JÜRGEN P. KdV方程和KAM理论[M]. 影印版. 北京:高等教育出版社,2010:19-39. LUO A C J. VALENTIN A. Hamiltonian chaos beyond the KAM theory[M]. Beijing:Higher Education Press, 2010:4-34. ARNOLD V I. Proof of a theorem of A. N. Kolmogorov on the invariance of quasi-periodic motions under small perturbations of the Hamiltonian[M].[S.1.]:Ross Mathematical Survey, 1963:9-36. MOSER J. On invariant curves of area-periodic mappings of an annulus[J]. Nachr. Akad. Wiss. Göttingen Math. Phys., 1962, KI.Ⅱ:1-20. 程崇庆,孙义燧. 哈密顿系统中的有序和无序运动[M]. 上海:上海科技出版社,1996:39-50. 李继彬,赵晓华,刘正荣. 广义哈密顿系统理论及其应用[M]. 北京:科学出版社,1994:178-195. 从福仲. KAM方法和系统的KAM稳定性[M]. 北京:科学出版社,2013:2-15. ARNOLD V I. Mathematical methods of classical mechanics[M]. New York:Springer-Verlag, 1987:163-188, 271-291. ARNOLD V I. Dynamical systems Ⅲ[M]. New York:Springer-Verlag, 1998:122-124. 胡志兴,管克英. 复杂双摆的KAM定理[J]. 高校应用数学学报:A辑,1999,14(2):147-154. HU Zhixing, GUAN Keying. KAM theory of the complex double pendulum[J]. Applied Mathematics A Journal of Chinese Universities, 1999, 14(2):147-154. 胡志兴,管克英. 陀螺仪运动的混沌与KAM理论[J]. 应用数学学报,2000,23(2):212-220. HU Zhixing, GUAN Keying. The chaos and KAM theory of the gyroscope[J]. Acta Mathematicae Applicatae Sinica, 2000, 23(2):212-220. 雷锦志,管克英. 圆型限制性三体问题中双恒星系统的存在性[J]. 高校应用数学学报:A辑,2001,16(1):55-60. LEI Jinzhi, GUAN Keying. Existence of double star system in circular restricted three-body problem[J]. Applied Mathematics A Journal of Chinese Universities, 2001, 16(1):55-60. -
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