Abstract—

The plane motion of a solid in a uniform gravity field is studied. The solid is suspended on a weightless inextensible thread, which remains stretched during the motion. It is assumed that the length of the thread is large (\(\sim {{\varepsilon }^{{ - 1/2}}}\)) and the distance from the point of solid suspension to its center of gravity is small (~ε). The equations of motion are presented as equations of a system with one rapidly rotating phase. This system is analyzed using classical perturbation theory and KAM theory. It is shown that for all values of time, the movement differs little (by ~ε) from the slow oscillations of the thread in the vicinity of the descending vertical and the solid rotation relative to the suspension point with an almost constant angular velocity. The measure of the set of motions, different from the above motions, is estimated from above by the value of the order of \(\exp \left( { - c{\text{/}}\varepsilon } \right)\) (\(c > 0\) = const).

中文翻译：

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均匀重力场中悬浮在螺纹上的固体的运动演化

摘要-

研究了均匀重力场中固体的平面运动。固体悬浮在一根失重且不可伸长的线上，在运动过程中保持拉伸。假设线程的长度很大（\(\sim {{\varepsilon }^{{ - 1/2}}}\)) 并且固体悬浮点到其重心的距离很小 (~ε)。运动方程表示为具有一个快速旋转相位的系统方程。该系统使用经典微扰理论和 KAM 理论进行分析。结果表明，对于所有时间值，运动与线在下降垂线附近的缓慢振荡和相对于悬挂点的固体旋转几乎没有差异（约 ε），角速度几乎恒定。一组运动的度量，与上述运动不同，是从上面通过\(\exp \left( { - c{\text{/}}\varepsilon } \right)\)阶的值估计的（\(c > 0\) = 常量）。