This paper takes the seventh question of the second round of the 33rd National High School Physics Competition as the research object and explores the dynamic behavior of the double pendulum system with equal mass and length. First, based on Lagrangian mechanics, it precisely solves the instantaneous angular acceleration and the force of interaction between the rods under specific initial conditions. Then, through numerical simulation, it analyzes the motion trajectory of the system under large-angle oscillation and reveals its typical chaotic characteristics. Finally, it systematically analyzes the dynamic response of the system near the equilibrium position when perturbed. By introducing the small-angle approximation and selecting appropriate independent modal coordinates, it successfully derives the approximate analytical solution for the system's small-amplitude vibration. The results show that under perturbation, the system exhibits regular ordered motion, with its dynamics being the linear superposition of two independent normal modes. This study not only provides a clear physical picture and theoretical support for understanding the transition between chaotic and ordered behavior in the double pendulum system, but also fills the gap in related teaching research with the perturbation analytical solution. It offers a vibration analysis case with both theoretical foundation and exploration value for physics competition teaching and theoretical mechanics courses.

A small ball can perform pure rolling, as well as rolling and sliding motion on a rough circular orbit. Using the theorem of center of mass motion and the theorem of angular momentum of the particle system, it is concluded that the motion of a small ball on a circular orbit is closely related to the dynamic friction coefficient. The motion of a small ball on a rough circular track is significantly different from that of a small slider on a rough circular track. The rotation of the small ball cannot be ignored and cannot be processed as a particle model. To avoid the influence of rotation, it is recommended to select the object as a small slider when setting the problem of controlling the movement of an object on a circular orbit in middle school.