两磁相互作用体系统轨道运动稳定性研究
Research into Orbital Motion Stability in System of Two Magnetically Interacting Bodies
摘要：
本文描述了两个只受磁力作用的长圆柱形磁体的轨道运动的稳定性。为了进行分析研究，使用了磁相互作用的对称顶部模型[1]。该模型以前是在电磁场的准静态方法中开发的，该方法基于相互作用的磁体能量的一般表达式[2]。在轨道运动稳定性的研究中，所谓的相对平衡[3]起着特殊的作用，即系统动力学的轨迹同时是系统不变性组的单参数子群。目前，人们通常采用两种类似的方法来研究它们的稳定性：能量-动量方法和能量-Casimir方法。在[4]的定理中给出了系统稳定性研究的最合适的判据;该稳定性判据成功地推广了上述两种方法，并涵盖了基于泊松结构的Hamilton形式主义[1]。由该定理导出了圆轨道稳定的充要条件。
英文摘要：
The stability of the orbital motion of two long cylindrical magnets interacting exclusively with magnetic forces is described. To carry out analytical studies a model of magnetically interacting symmetric tops [1] is used. The model was previously developed within the quasi-stationary approach for an electromagnetic field based on the general expression of the energy of interacting magnetic bodies [2]. A special role in the investigation of the stability of orbital motions is played by the so-called relative equilibria [3], i.e. the trajectories of the system dynamics which are at the same time one-parameter subgroups of the system invariance group. Nowadays their stability is normally investigated using two similar approaches -- energy-momentum and energy-Casimir methods. The most suitable criterion for the system stability investigation was formulated in the theorem of [4]; this stability criterion successfully generalizes both the methods mentioned above and covers the Hamiltonian formalism based on Poisson structures [1]. The necessary and sufficient conditions for the circular orbit stability are derived from this theorem.
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