Reduced mass

If move two bodies with masses and under the influence of a vanishing total force, so can the equations of motion in the free movement of the center of gravity and the one- body problem of the relative movement split. Here, the relative distance as a particle that has the reduced mass moved

Depending on the mass of the heavier body has the reduced mass values ​​between and. In important cases ( planetary motion, motion of an electron in the Coulomb field of the atomic nucleus ), the masses of the heavier and the lighter body several orders of magnitude different. Then, the reduced mass is almost the mass of the lightest particle,

In many textbooks the reduced mass by the Greek letter μ, sometimes with or is abbreviated.

Derivation

For vanishing total force the equations of motion for the places and the two bodies be

Adding up both equations, we obtain for the focus

The equation of motion

Of a free particle. So, the focus moves in a straight line, uniform,

Dividing the equation of motion of the particles by the mass and receives the difference one obtains the equation of motion for the spacing

It therefore moves like a particle of reduced mass under the influence of the force.

• Theoretical Mechanics