导电弹性杆在磁场中的可积性
Integrability of a conducting elastic rod in a magnetic field
摘要：
本文考虑了由电动空间系绳问题引起的、置于均匀磁场中的导电弹性杆的平衡方程。当表示在身体坐标的方程被发现坐在一个层次的非正则哈密顿系统涉及越来越多的向量场。这些系统，其中包括经典的欧拉和基尔霍夫杆，被证明是完全可积的横观各向同性杆的情况下，他们实际上是由一个拉克斯对。对于磁棒，这给出了以前提出的抽象九维可积系统的物理解释。我们使用的守恒量减少到一个四维正则哈密顿系统的方程，允许通过庞加莱截面相空间的几何形状进行调查。在特殊情况下，当杆中的力与磁场一致时，系统是超可积的，这意味着相空间完全分解为周期轨道，对应于直的扭曲杆。
英文摘要：
We consider the equilibrium equations for a conducting elastic rod placed in a uniform magnetic field, motivated by the problem of electrodynamic space tethers. When expressed in body coordinates the equations are found to sit in a hierarchy of non-canonical Hamiltonian systems involving an increasing number of vector fields. These systems, which include the classical Euler and Kirchhoff rods, are shown to be completely integrable in the case of a transversely isotropic rod; they are in fact generated by a Lax pair. For the magnetic rod this gives a physical interpretation to a previously proposed abstract nine-dimensional integrable system. We use the conserved quantities to reduce the equations to a four-dimensional canonical Hamiltonian system, allowing the geometry of the phase space to be investigated through Poincaré sections. In the special case where the force in the rod is aligned with the magnetic field the system turns out to be superintegrable, meaning that the phase space breaks down completely into periodic orbits, corresponding to straight twisted rods.
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