# Proving Newtons third law

1. Aug 13, 2012

### aaaa202

Sometimes I wonder if it actually possible to prove the action-reaction principle. I do know that Newtons laws are empirical but then if you think about the third law it says that in interaction between two bodies they exert equal and opposite forces on each other. Now what happens in the collision between two classical bodies is purely determined by the laws of electromagnetism and in Coulombs law, the action reaction principle is built in directly. The same can be said about interactions which involves gravity - here the action-reaction principle is also just a purely mathematical thing in the law of gravitation.
So was Newtons observation of action-reaction not just a manifestation of these two laws?

2. Aug 13, 2012

### Nessdude14

Newton's third law is evident in almost any situation in classical mechanics. You can pick any force on a free body diagram for an object, and you will find a force on another object which is exactly equal in magnitude and opposite in direction. You can do countless experiments to confirm that this force exists. In fact, practically any experiment involving classical mechanics can be used as evidence for Newton's 3rd law.

Here's a situation that might give you an intuitive idea of Newton's 3rd law: When you press on a wall, you exert a force on the wall (although it probably won't move). You can also feel the wall pressing back, exerting a force on your hand. This force from the wall on your hand is the 3rd-law pair to the force that you're applying to the wall. Whenever you push on something, it pushes back at you with the exact same force, in accordance with Newton's 3rd law.

3. Aug 13, 2012

### Staff: Mentor

The power of Newton's third law comes from its simplicity and generality; it applies to all interactions between masses, regardless of the details of the interaction. It's easy enough to see the equal and opposite forces at work in two-body gravitational and/or electromagnetic interactions (although it would be well to remember that Newton's third law inspired the mathematical formulation of Coulomb's law and gravitation - neither is especially obvious if you aren't already thinking in third-law terms).

But suppose you had to analyze a more complex situation, like say for example the motion of an inclined plane that is floating in water with a helicopter hovering above it? It is enormously helpful to be able to invoke the third law up front, to know without grovelling through all the fluid dynamics, that the downforce on the plane is equal to the upforce on the helicopter.

4. Aug 13, 2012

### zoobyshoe

Newton gives a proof:

http://archive.org/stream/newtonspmathema00newtrich#page/n97/mode/2up

Look at the text around the diagram of the circle cut into segments starting with the last paragraph on page 92. The circle represents the earth.

His proof boils down to saying that, if the third law were not true we'd see very different results in all cases. Eg: If the reaction did not equal the action, the action would dominate causing infinite acceleration. "Which", Newton asserts, "is absurd". And also violates the first law.

5. Aug 13, 2012

### atyy

Yes, if the underlying fundamental laws were Coulomb's law or Newton's law of universal gravitation, then the third law can be derived from them.

However, if we don't know all the fundamental laws, then Newton's third law says that we at least know a symmetry of these fundamental laws.

The symmetry of Newton's third law is equivalent to momentum conservation.

Although there is no "direct" generalization of Newton's third law to special relativity, the conservation of momentum does generalize "directly" to special relativity.