Consider two particles at positions r₁ and r₂, attracted by a force that depends on the distance

between them. The kinetic energy of the system is

and the potential energy U is a function of the distance r. The Lagrangian is therefore

We want to simplify the problem by removing the motion of the center of mass of the system. In a system where there are no external forces acting on the particles, the center of mass of the system,

moves with a constant velocity, independent of the motions of the particles. Instead of writing the Lagrangian as a function of the positions r₁ and r₂ of the particles, we will write it as a function of the position of the center of mass R and the particle separation

In terms of these variables, the positions of the particles are

r₁ =

r₂ =

and the Lagrangian is

The Lagrangian is a sum of terms containing only and terms containing only . This means that and don't appear in each others' equations of motion. A Lagrangian with this property is said to be separable, and can be written as a sum of two Lagrangians each governing part of the system. From (7), we see that the equation of motion for is

The center of motion is not accelerating, which confirms that momentum is conserved in this system.