How Newton's laws leads to law of conservation of momentum?
In the following, Newton’s second law is regarded as given.
Conservation of linear momentum is a consequence of the third law.
Imagine two bodies interacting with each other and forming an isolated system. The change in momentum of one body is the time integral of the force on it. By the third law, the forces are equal and opposite so the two time integrals and the changes in momenta have the same magnitude but opposite signs. That is the sum of the changes in momenta is zero.
It is easy to generalize the above derivation to any number of particles or bodies. I.e., given the second law, Newton’s third law implies conservation of momentum.
Conservation of angular momentum is a consequence of the third law.
The third law as stated by Newton is not just that the forces are equal and opposite but that they also have the same line of action. That action and reaction have the same line of action implies conservation of angular momentum (for isolated systems). This is quite easy to show.
Conservation of linear and angular momentum imply the third law.
If we are given that conservation of momentum holds for all systems and that forces occur in pairs, a formal proof is not difficult.
The significance of the third law.
In addition to the above, the universe would behave very differently if the third law did not hold. From the observed behavior of our cosmos we can conclude that the third law holds as at least and excellent approximation.
This shows that Newton’s third law is fundamental and not just an afterthought. Even in electromagnetism, where the third law does not hold for particles, it does hold if field momenta are accounted for.
In the following, Newton’s second law is regarded as given.
Conservation of linear momentum is a consequence of the third law.
Imagine two bodies interacting with each other and forming an isolated system. The change in momentum of one body is the time integral of the force on it. By the third law, the forces are equal and opposite so the two time integrals and the changes in momenta have the same magnitude but opposite signs. That is the sum of the changes in momenta is zero.
It is easy to generalize the above derivation to any number of particles or bodies. I.e., given the second law, Newton’s third law implies conservation of momentum.
Conservation of angular momentum is a consequence of the third law.
The third law as stated by Newton is not just that the forces are equal and opposite but that they also have the same line of action. That action and reaction have the same line of action implies conservation of angular momentum (for isolated systems). This is quite easy to show.
Conservation of linear and angular momentum imply the third law.
If we are given that conservation of momentum holds for all systems and that forces occur in pairs, a formal proof is not difficult.
The significance of the third law.
In addition to the above, the universe would behave very differently if the third law did not hold. From the observed behavior of our cosmos we can conclude that the third law holds as at least and excellent approximation.
This shows that Newton’s third law is fundamental and not just an afterthought. Even in electromagnetism, where the third law does not hold for particles, it does hold if field momenta are accounted for.
