Based on the Hamilton principle,a numerical algorithm associated with Lagrange multiplier method for the classical motion path problem is proposed. Different from traditional variational method,the present algorithm transforms the classical motion path problem into the conditional extremum problem with respect to the motion equation. By utilizing this algorithm,numerical solutions to motion path problems in one-dimensional gravitation potential and one-dimensional elastic potential are obtained and compared with the corresponding analytical results,respectively. Such two examples can be used as practical and interesting teaching cases for the relevant curriculums,e.g. Mechanics in college physics,Hamilton principle in theoretical mechanics and numerical calculation in computational mathematics. These examples are helpful for students to understand the Hamilton principle more deeply,and improve the ability of applying the knowledge in the fields of physics,mathematics, and computer science.%基于哈密顿原理，提出经典运动路径问题的拉格朗日乘数数值算法。与传统的变分方法不同，该算法将经典运动路径问题改写为关于路径方程的条件极值问题。利用该算法得到了一维重力势中的运动路径和一维弹性势中的运动路径的数值解，并与各自的解析解作了比较分析。这2个例子可以作为大学物理力学、理论力学哈密顿原理以及计算数学数值计算等相关课程内容实用教学案例，其有助于学生更深刻地理解哈密顿原理，提高综合应用物理、数学、计算机科学等知识的能力。
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