llcotp.com/LL1/13_problem.md

PROBLEM A system consists of one particle of mass M and n particles with equal masses m. Eliminate the motion of the centre of mass and so reduce the problem to one involving n particles. SOLUTION. Let R be the radius vector of the particle of mass M, and Ra (a = 1, 2, ..., n) those of the particles of mass m. We put ra = Ra-R and take the origin to be at the centre of mass: MR+mER = 0. Hence where =M + nm; Ra = R + ra. Substitution in the Lagrangian L = gives ra. The potential energy depends only on the distances between the particles, and so can be written as a function of the ra.