majidstar1234/Path_Space_Topology_Project: Path Space Topology and Dissipation-Induced Uniqueness in a Simple Conservative System

Description:This project explores a fundamental question in variational mechanics: Why does a small amount of friction make a physical path unique? Using a 1-DOF pendulum-like system, we provide numerical evidence that the "Principle of Least Action" becomes degenerate near saddle points.Our key contribution is the use of Hessian Spectral Analysis to show that while conservative systems have a "flat" action landscape (zero eigenvalues), adding linear damping ($b > 0$) opens a Spectral Gap. This gap acts as a topological anchor, selecting a unique, physically consistent trajectory from an infinite set of mathematical possibilities.