Abstract: This paper compares the traditional numerical methods for equations of motion with recently developed finite element (FE) methods. The comparison is conducted on aspects such as stability conditions, nodal truncation errors, convergence orders for nodal solutions, and period elongation. Relevant formulas are provided for each aspect. The comparative analysis demonstrates that the recently proposed condensed element and reduced element based on the first-order equations of motion perform well, with superior performance in various aspects. Particularly, the first-order reduced element is a high performance adaptive time-stepping element with a built-in maximum norm error estimator.