Past exam of the mathematics course of the University of Cambridge / 2015 / ii / Paper 4 / 7D / a / Solution

Let $x_1,x_2,x_3$ be longitudinal displacements from equilibrium. The original diagram has masses $m,2m,m$, and no attachments to fixed walls. The Lagrangian isThe Euler-Lagrange equations giveThe forces sum to zero, consistent with momentum conservation for an isolated molecule.