In a closed system, how does the Principle of Least Action relate to the motion of a particle under conservative forces?

The Principle of Least Action says that a particle moving in a closed system follows the path that makes the integral of its Lagrangian—the difference between kinetic and potential energy—stationary. By setting the variation of this action to zero, we derive the Euler–Lagrange equations, which are mathematically equivalent to Newton’s second law for conservative forces. Thus, the particle’s motion is determined by the same equations that describe forces derived from a potential energy function. For example, a ball thrown upward in a gravitational field follows a parabolic trajectory that satisfies the least‑action condition, leading to the familiar \(y(t)=y_0+v_0t-\frac12gt^2\) path. This shows that the least‑action principle is just a compact way to recover the familiar equations of motion for conservative forces.