关键词: 角动量守恒定律, 机械能守恒定律, 欧拉-拉格朗日方程, 哈密顿正则方程, 广义相对论的弱场近似理论

Abstract: 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.

Key words: conservation of angular momentum, conservation of mechanical energy, Euler-Lagrange equation, Hamilton’s canonical equations, weak field approximation of general relativity