  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.
