We propose a constrained high-index saddle dynamics (CHiSD) to search index-$k$ saddle points of an energy functional subject to equality constraints. With Riemannian manifold tools, the CHiSD is derived in a minimax framework, and its linear stability at the index-$k$ saddle point is proved. To ensure the manifold properties, the CHiSD is numerically implemented using retractions and vector transport. Then we present a numerical approach by combining CHiSD with downward and upward search algorithms to construct the solution landscape with equality constraints. We apply the Thomson problem and the Bose--Einstein condensation as the numerical examples to demonstrate the efficiency of the proposed method.

翻译：我们建议使用一个有限的高指数马鞍动态(CHISD)来搜索受平等制约的能源功能的指数-美元马鞍点。用里曼尼多种工具,CHISD从一个迷你马克斯框架中衍生出来,并且证明了其在指数-美元马鞍点的线性稳定性。为了确保多重特性,CHISD使用撤回和矢量传输进行数字化实施。然后我们提出一个数字方法,将CHISD与下向和上向搜索算法结合起来,在平等制约下向和向上构建解决方案。我们用Thomson问题和Bose-Einstein Condencensation作为数字例子来展示拟议方法的效率。