The local Morse Homology of the critical points in the Lagrange problem

In this paper, we construct local Morse homology in a new way and compute the local Morse homology of the critical points of the Lagrange problem. As a corollary, we prove for the first time that each of the linear critical points is either a saddle point or a degenerate critical point compared to the previous conclusion that if all the linear critical points are non-degenerate, they are saddle points.