Abstract

We consider the dynamics of a rod on the plane in a flow of non-interacting point particles moving at a fixed speed. When colliding with the rod, the particles are reflected elastically and then leave the plane of motion of the rod and do not interact with it. A thin unbending weightless “knitting needle” is fastened to the massive rod. The needle is attached to an anchor point and can rotate freely about it. The particles do not interact with the needle.

The equations of dynamics are obtained, which are piecewise analytic: the phase space is divided into four regions where the analytic formulas are different. There are two fixed points of the system, corresponding to the position of the rod parallel to the flow velocity, with the anchor point at the front and the back. It is found that the former point is topologically a stable focus, and the latter is topologically a saddle. A qualitative description of the phase portrait of the system is obtained.

中文翻译：

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稀薄流中的摆动力学

摘要

我们考虑平面上杆在以固定速度移动的非相互作用点粒子流中的动力学。当与杆碰撞时，粒子被弹性反射，然后离开杆的运动平面并且不与其相互作用。一根细长的、不弯曲的、失重的“织针”固定在粗大的杆上。针连接到锚点并可以绕其自由旋转。颗粒不与针相互作用。

得到了分段解析的动力学方程：将相空间分为四个区域，其中解析公式不同。系统有两个固定点，对应于杆与流速平行的位置，锚点位于前部和后部。发现前一个点在拓扑上是稳定焦点，而后者在拓扑上是鞍点。获得系统相图的定性描述。