Vol 1 - L D Landau, E.M. Lifshitz - Mechanics · Problem 13.1

The key is recognizing that the centre-of-mass motion decouples from the internal dynamics when you work in the centre-of-mass frame. By shifting to coordinates that measure positions relative to the centre of mass, you remove the three translational degrees of freedom, leaving only the $n$ relative coordinates that describe how the particles move with respect to each other. This transforms the problem from $n+1$ particles in the lab frame into an equivalent system of $n$ particles in the centre-of-mass frame, with an effective reduced-mass structure.

The key is recognizing that the centre-of-mass motion decouples from the internal dynamics when you work in the centre-of-mass frame. By shifting to coordinates that measure positions relative to the centre of mass, you remove the three translational degrees of freedom, leaving only the $n$ relative coordinates that describe how the particles move with respect to each other. This transforms the problem from $n+1$ particles in the lab frame into an equivalent system of $n$ particles in the centre-of-mass frame, with an effective reduced-mass structure.