Worked example - two coupled pendulums. Two identical pendulums (mass m, length l) joined by a spring k at the bobs; small angles x1, x2. T = (1/2) m (x1_dot^2 + x2_dot^2); V = (1/2)(mg/l)(x1^2 + x2^2) + (1/2) k (x1 - x2)^2. Equations: m x1_ddot = -(mg/l) x1 - k(x1 - x2), m x2_ddot = -(mg/l) x2 + k(x1 - x2). Try x = a cos(omega t): the secular determinant gives omega1^2 = g/l (in-phase mode: a1 = a2, spring never stretches) and omega2^2 = g/l + 2k/m (anti-phase: a1 = -a2). Start only pendulum 1: the motion is the sum of both modes, producing beats - energy transfers fully to pendulum 2 and back with beat frequency (omega2 - omega1)/2. This is classical energy exchange, the analogue of quantum tunnelling oscillations.