Lyapunov energy of the accelerated flow (convex)
The accelerated gradient flow \ddot X + \tfrac{3}{t}\dot X + \nabla f(X)=0 reaches the \mathcal{O}(1/t^2) rate. The proof uses the Lyapunov energy E(t)=t^2\bigl(f(X(t))-f^*\bigr)+2\Bigl\|X(t)-x^*+\tfrac{t}{2}\dot X(t)\Bigr\|^2, which is non-increasing along the trajectory. Since E(t)\le E(0)=2\|x_0-x^*\|^2, the first term gives f(X(t))-f^*\le 2\|x_0-x^*\|^2/t^2. The objective oscillates under the curve while the energy decreases monotonically.