从一道中学生物理竞赛题出发，深入研究了重力作用下滑块在光滑半球面上的运动问题. 在采用最简单的守恒定律方法解答原题设问的基础上，应用牛顿力学通过对滑块进行受力分析，推导出了滑块的动力学方程. 在此基础上从拉格朗日力学、哈密顿力学和引力的弱场近似理论的视角对该问题进行深入探究，推导出了与牛顿力学方法所得结果一致的动力学方程. 随后，对得到的动力学方程进行数值求解，绘制了滑块的运动轨迹. 最后，将重力作用下滑块在光滑半球面与竖直光滑圆锥面上的运动轨迹进行对比，分析了两者的异同.

Starting from a mechanics problem of Chinese Physics Olympiad, this paper delves into the motion of a slider on a smooth hemisphere under the influence of gravity. Beginning with a straightforward application of conservation laws to address the original query, this paper proceeds to derive the slider’s dynamic equations through force analysis using Newtonian mechanics. Furthermore, an analysis is carried out from the perspectives of Lagrangian and Hamiltonian mechanics, as well as the weak field approximation of general relativity, deriving dynamic equations consistent with those obtained via Newtonian mechanics. Numerical solutions are then employed to solve the dynamic equations, allowing for the visualization of the slider’s trajectory. Finally, a comparison is made between the trajectory of the slider’s motion on the smooth hemisphere under gravity and that on a vertical smooth conical surface.