B.5.TC:5.2 - The same oscillator, different formulation and computation

Consider an ideal one-dimensional mass on a linear spring, with mass m > 0, spring constant k > 0, displacement q and velocity v. Friction and external forcing are absent. The question is its motion from q(0) = a and v(0) = 0.

The force account gives m q’’ = -k q. The variational account uses L(q,v) = m v²/2 - k q²/2. Its Euler-Lagrange equation is d/dt(∂L/∂v) - ∂L/∂q = 0; substitution gives m q’’ + k q = 0. Both therefore give q(t) = a cos(√(k/m) t) under these premises. For this motion question, the two constructions agree. The variational formulation can become preferable when another coordinate choice simplifies constraints; the force formulation may be the quicker account for this simple case.

Suppose a computation appears to disagree. Take m = k = a = 1 and use the forward Euler updates q_next = q + h v and v_next = v - h q. From q = 1, v = 0 and h = 0.1, it returns q_next = 1, v_next = -0.1. The energy (q² + v²)/2 rises from 0.5 to 0.505. In fact these updates multiply energy by 1 + h² at each step, whereas the differential equation conserves it.

The discrepancy is produced by the approximation used in the computation. Compare an adequate step size or another numerical method under C.29.2 for the requested horizon and error. If the real oscillator loses energy, examine friction and other physical interactions; that changes the model question. The equality of the two ideal derivations remains available within its premises.