Kirchhoff相似性原理與彈性纖體的空間混沌構形

Abstract

著重討論三維彈性纖體空。通過研究陀螺體的受擾姿態運動的哈密頓結構及與其相關的Melnikov積分, 解析地確定了彈性纖體靜態混沌構形可能產生的條件。採用7～8階Runge-Kutta算法定量地核對了由Melnikov方法所得的定性結果。三維彈性纖體存在於不同尺度的物質結構中(從微觀的DNA雙螺旋結構到宏觀的彈性細桿、細繩、電纜、音像磁帶和衛星系繩等)。仿真結果表明, 在合適的載荷條件下, 彈性纖體的平衡構形將呈現起因於同宿/異宿分叉的混沌。

The aim of this paper is to discuss the extended Kirchhoff analogy between the spatial equilibrium of a 3D force-free buckled elastica and the temporal dynamics of the torque-free gyrostat. In conjunction with the Melnikov integral, we shall determine analytically the conditions for the possible onset of spatial chaos in the elastica by exploring the Hamiltonian structure of the rotational motion of a perturbed gyrostat. The qualitative results are quantitatively cross-checked by the 7 ～ 8th order Runge-Kutta algorithm. The elastica appears at different scales from microscopic chains of super coiling DNA structures to macroscopic rods/ropes/cables/satellite tethers. The simulation results show that there exists homoclinic/heteroclinic bifurcations to chaos in the equilibrium of the elastica under the appropriate load conditions, equivalently, boundary conditions.