High-index saddle dynamics (HiSD) serves as a competitive instrument in searching the any-index saddle points and constructing the solution landscape of complex systems. The Lagrangian multiplier terms in HiSD ensure the Stiefel manifold constraint, which, however, are dropped in the commonly-used discrete HiSD scheme and are replaced by an additional Gram-Schmidt orthonormalization. Though this scheme has been successfully applied in various fields, it is still unclear why the above modification does not affect its effectiveness. We recover the same form as HiSD from this scheme, which not only leads to error estimates naturally, but indicates that the mechanism of Stiefel manifold preservation by Lagrangian multiplier terms in HiSD is nearly a Gram-Schmidt process (such that the above modification is appropriate). The developed methods are further extended to analyze the more complicated constrained HiSD on high-dimensional sphere, which reveals more mechanisms of the constrained HiSD in preserving several manifold properties.

翻译：高阶鞍点动力学（HiSD）是搜索任意阶鞍点及构建复杂系统解景观的有效工具。HiSD中的拉格朗日乘子项确保施蒂费尔流形约束，但在常用离散化HiSD方案中，这些项被省略，并由额外的Gram-Schmidt正交化过程替代。尽管该方案已成功应用于多个领域，但上述修改为何不影响其有效性仍不明确。我们从该方案中恢复了与HiSD相同的形式，这不仅自然推导出误差估计，还表明HiSD中通过拉格朗日乘子项保持施蒂费尔流形的机制近似于Gram-Schmidt过程（因此上述修改是合理的）。所发展的方法进一步扩展至分析高维球面上更复杂的约束HiSD，揭示了约束HiSD在保持多种流形性质中的更多机制。