Noether energy conservation: H = kx²/2 + mẋ²/2 = const, dH/dt ≡ 0

Explicit Noether-theorem conservation of energy for the harmonic-oscillator Lagrangian. The Lagrangian L = m·ẋ²/2 − k·x²/2 has no explicit time dependence (∂L/∂t = 0), so time-translation invariance yields energy conservation: H =…