14.5: Reduced Mass Coordinates: Isolated Two-body Problem

In classical mechanics, the two-body problem is one of the fundamental problems describing the motion of two interacting bodies under gravity or any other central force. When considering the motion of two bodies, one of the most important concepts is the reduced mass coordinates, a quantity that allows the two-body problem to be solved like a single-body problem. In these circumstances, it is assumed that a single body with reduced mass revolves around another body fixed in a position with an infinite mass. The reduced mass of such a system is given by

Consider a two-particle system of different masses at distances r₁ and r₂ from the origin. The center of mass of this system lies on the line joining the two particles and can be expressed as

If the center of mass is considered the origin, it divides the line joining the two particles in the inverse ratio of their masses. When no external force is acting on the system of the two particles, and the only forces are mutual interactions, the system behaves as a single particle with a reduced mass. It acts at the center of the mass of the two particles. The importance of the center of mass and reduced mass lies in the fact that, instead of having two separate equations of motion, a single equation of motion involving reduced mass reduces a two-body problem to a one-body problem. In conclusion, reduced mass coordinates are a powerful tool in studying the two-body problem, particularly in the isolated two-body problem.

The earth rotates around the sun. According to Newton's law of gravitation, the sun and earth exert equal and opposite forces on each other. So why does only the earth revolve?

Consider two bodies moving at different velocities. The velocity of the center of mass of this system lies on the line joining their individual velocities.

If the system is isolated, the center of mass can be taken as the origin. The velocities of the bodies can now be expressed relative to the velocity of the center of mass.

Multiplying the velocities with their respective masses gives their momenta. The first term here denotes the reduced mass of the two-body system.

The equation of the two-body system is now converted into a one-body system.

In the earth-sun system, the center of mass velocity can be approximated to the velocity of the sun. So, the sun has negligible relative velocity with respect to its center of mass, whereas the earth possesses a relative velocity which allows it to revolve around the sun.