Newton's Third Law of Motion

One corollary of Newton's third law is that an object cannot exert a force on itself. Another corollary is that all forces in the Universe have corresponding reactions. The only exceptions to this rule are the fictitious forces which arise in non-inertial reference frames (

It should be noted that Newton's third law implies *action at a
distance*. In other words, if the force that object
exerts on object suddenly changes then Newton's third law
demands that there must be an *immediate*
change in the force that object
exerts on object . Moreover, this must be true irrespective of the
distance between the two objects. However, we now know that
Einstein's theory of relativity forbids information from traveling through the
Universe faster than the velocity of light in vacuum. Hence, action at a distance is also forbidden. In other words, if the force that object
exerts on object suddenly changes then there must be a
*time delay*, which is at least as long as it takes a light ray to propagate
between the two objects, before the force that object
exerts on object can respond. Of course, this means that
Newton's third law is not, strictly speaking, correct. However, as
long as we restrict our investigations to the motions of dynamical
systems on time-scales that are long compared to the time
required for light-rays to traverse these systems, Newton's third
law can be regarded as being approximately correct.

In an inertial frame, Newton's second law of motion applied to the th object yields

(25) |

where is the total mass. The quantity is the vector displacement of the

According to Equation (26), the center of mass of the system moves in a uniform straight-line, in accordance with Newton's first law of motion, irrespective of the nature of the forces acting between the various components of the system.

Now, if the center of mass moves in a uniform straight-line then
the center of mass velocity,

A comparison of Equations (28) and (29) suggests that is also a constant of the motion. In other words, the total momentum of the system is a

Taking the vector product of Equation (24) with the position vector , we
obtain

(31) |

(32) |

(33) |

Consider the sum on the right-hand side of the above equation. A general term,
, in this sum can always be paired with a
matching term,
, in which the indices have been swapped.
Making use of Equation (23), the sum of a general matched pair can be written

(35) |

(36) |

In other words, the total angular momentum of the system is a