Lagrangian Mechanics
Find motion by following energy through a system.
Endpoints are fixed. Every candidate path starts and ends at the same positions and times; only the route between them varies.
Simplified: a unit mass on a unit spring (m = k = 1), so T = ½q̇² and V = ½q². Every candidate starts at q = 0 and ends at q = sin 2 when t = 2; varied paths add ε·sin(πt/2). Over this short time the true path is a minimum of S; over longer times it can be a saddle, which is why the principle says “stationary”.
This varied path keeps both endpoints but raises the action by 0.183 ≈ 0.734·ε². First-order changes cancel; that is what “stationary” means. Its T + V also drifts by 0.598, so Newton's laws would not allow it.
Make the idea yours.
Use the model, then answer without looking back. Each check isolates one decision a learner should be able to make after this lesson.