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

TL;DR

This paper introduces a novel approach to local Morse homology and applies it to analyze critical points in the Lagrange problem, revealing their saddle or degenerate nature.

Contribution

It develops a new method for constructing local Morse homology and characterizes the critical points of the Lagrange problem as saddle or degenerate.

Findings

All linear critical points are either saddle or degenerate.

First proof that linear critical points are saddle or degenerate.

New technique for local Morse homology construction.

Abstract

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.