Principle of Least Action: Path Relaxation

Mathematical physics rebuilds mechanics on a single idea: of all conceivable paths between two fixed events, nature follows the one that extremizes the action functional S[q] = ∫L dt. This simulator makes that abstract statement tangible — a trial trajectory is chopped into free nodes, randomly perturbed into a jagged 3D squiggle, and then relaxed frame by frame via gradient descent directly on the discretized action, with no equation of motion hard-coded anywhere. Watch the vertical coordinate bow into the exact gravitational parabola predicted by the Euler–Lagrange equation ÿ = −g while the unforced horizontal coordinates straighten into perfect lines, with live readouts for the action value, gradient-step count, and RMS deviation from the closed-form exact solution.