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Step 1: Choose a system – a mass m attached to a spring with spring constant k. The only coordinate we need is the displacement x from the spring’s rest position. Step 2: Write the kinetic energy T = (1/2) m ẋ². The kinetic energy depends on how fast the mass moves. Step 3: Write the potential energy V = (1/2) k x². The spring stores energy when it’s stretched or compressed. Step 4: Form the Lagrangian L = T - V = (1/2) m ẋ² - (1/2) k x². Step 5: Compute the partial derivatives: ∂L/∂ẋ = m ẋ ∂L/∂x = k x Step 6: Take the time derivative of ∂L/∂ẋ: d/dt (∂L/∂ẋ) = m ẍ. Step 7: Plug into the Euler‑Lagrange equation: m ẍ - k x = 0. Step 8: Rearrange to get the familiar equation of motion: m ẍ + k x = 0, which describes simple harmonic motion.