On the "origin" of the $3^\mathrm{rd}$ law of motion

Question

If two bodies exert forces on each other, these forces are equal in magnitude and opposite in direction.

This law is often addressed as the action-reaction principle, though the term "principle" might suggest we're dealing with a postulate. As such, it cannot be derived from more basic laws.

Since the "principle" of inertia has nothing dogmatic - it's just a particular case of the $2^{\rm nd}$ law, where the net force is zero - I thought the same could be said about the $3^{\rm rd}$ law.

A recurring example is that of the apple and the Earth.

An apple in free fall accelerates toward the Earth with a free fall acceleration. The force of the apple on the Earth also causes the Earth to accelerate toward the falling apple. By Newton's Third Law, the force of the Earth on the apple is exactly equal and opposite to the force of the apple on the Earth. (...)

The thing is... why does it have to be like this? This is what I've thought, but I'm not sure if it's a fair way of looking at it, nor if it can be used in general.

Consider the apple-Earth system, with the apple still attached to its tree.

Nothing is moving, and since there are no external forces acting on it (forget the Sun, the Moon, the other planets, etc. for a moment), the net force at any instant should be zero. Therefore, if it gains some internal forces at some point, like the weight force of the now falling apple, they must balance one another, otherwise an internal non-zero net force would make the whole system accelerate.

So.. qed?

I don't know. Is this approach valid in general? For example, are all the internal forces in a (not-accelerating) cloud balanced?

Answer 1

As you mentioned, the third law is just a postulate in classical mechanics. As a side note, a professor of mine would disagree with you that the first law is without independent content of the second law (although $a=0$ when $F=0$ is a special case of $F=ma$).

Nothing is moving, and since there are no external forces acting on it (forget the Sun, the Moon, the other planets, etc. for a moment), the net force at any instant should be zero. Therefore, if it gains some internal forces at some point, like the weight force of the now falling apple, they must balance one another, otherwise an internal non-zero net force would make the whole system accelerate.

This is an extremely insightful observation, but also a reductio ad absurdum. It's true that it would be absurd for a system to spontaneously accelerate itself, but this is not excluded by Newton's first two laws on their own. You have basically rediscovered the concept of the conservation of momentum, which reiterates your point that, if there are no external forces, the center of mass cannot accelerate.

There is also the seemingly obvious but really nontrivial fact that the center of mass of a system obeys $\vec{F}_{\mathrm{tot}}=M_{\mathrm{tot}}a_{\mathrm{cm}}$. This is the fundamental reason why things like tennis balls also follow Newton's second law even though they are really composed of small particles which are the only things guaranteed directly by Newton's laws to obey $F=ma$. This fact is usually just assumed, but would not be generally true if we don't include Newton's third law.

I think the fact that $F=ma$ ought to be "inherited" by big objects as a property gives some kind of elegant intuition for why the third law needs to be included as a postulate.

Answer 2

Newton's Third Law of motion is a direct consequence of the law of conservation of momentum. In this respect, Newton's third law is more fundamental than seems to be implied. If you change the momentum on one object, then some other object must change its momentum (in an opposite direction), so that the total momentum is conserved. Since forces cause a (rate in) change in momentum, this means that every force must have an equal but opposite reaction force.

Nothing is moving, and since there are no external forces acting on it (forget the Sun, the Moon, the other planets, etc. for a moment), the net force at any instant should be zero

The sum of the forces acting on the apple are zero. There are forces still acting on it, except in this case the forces are in equilibrium and there is no acceleration.

There is a tension force on the branch holding the apple preventing it from falling. In other words, the gravitational force on the apple due to the earth equals the tension force in the branch.

The apple still attracts the earth and the earth still attracts the apple. There does not need to be an acceleration for both the forces to be acting. If the forces are unbalanced then there will be an acceleration.

Therefore, if it gains some internal forces at some point, like the weight force of the now falling apple, they must balance one another, otherwise an internal non-zero net force would make the whole system accelerate.

The apple still has a weight force (by "internal" you mean "external") before it falls and the forces are in equilibrium, or $$T-W_a=0$$ where $T$ is tension and $W_a$ is the apple's weight.

For example, are all the internal forces in a (not-accelerating) cloud balanced?

Yes, but you mean external forces. If the cloud is not accelerating, its weight and its buoyant force are in equilibrium or $$F_B-W_a=0$$

Answer 3

Newton's 3rd Law is no longer merely a postulate but follows mathematically from the symmetry and homogeneity of space, i.e., no preferred location. We don't think that Newton knew this reason, but he was an intuitive genius, and he hypothesized this correctly. How does symmetry lead to the result of Newton's 3rd Law?

Linear momentum is conserved. It is neither created nor destroyed, but is transferred from one object to another. This, according to Emmy Noether's theorem about symmetry and conservation, is a consequence of the symmetry of configuration space. Expressed mathematically, $$\vec{p}_{\mathrm{final}}=\vec{p}_{\mathrm{initial}}+\vec{J}$$ where $\vec{J}$ is the current of momentum for the object or system being described: $$\vec{J}=\int_{t_i}^{t_f} \vec{F} \mathrm d t$$ where $\vec{F}$ is the net force acting $on$ the object or system.

Consider two particles interacting with each other (only). By interacting, I mean that particle 1 exerts a force on particle 2 , $\vec{F}_{21}$ (read this as "force on 2 by 1". And possibly 2 exerts a force on 1

Because there is no other force acting on 2 we can say $$\vec{p}_{2f}=\vec{p}_{2i}+\vec{J_{21}}$$

We can use the same reasoning for particle 1 and write $$\vec{p}_{1f}=\vec{p}_{1i}+\vec{J_{12}}$$

Now because there are no forces from outside the system, the net impulse on the system must be zero, and using the conservation of momentum continuity equation we get $$\vec{p}_{f}=\vec{p}_{i}+0 \\ \vec{p}_{1f}+\vec{p}_{2f}=\vec{p}_{1i}+\vec{p}_{2i}+\vec{J}_{12}+\vec{J}_{21} \\ \vec{J}_{12}=-\vec{J}_{21}$$

Because the time interval of interaction must be identical ($t_i \to t_f$), that means the forces must be equal and opposite. Neither force starts or quits before the other: $$\int_{t_i}^{t_f} \vec{F}_{12} \mathrm d t = -\int_{t_i}^{t_f} \vec{F}_{21} \mathrm d t \\ \vec{F}_{12}=-\vec{F}_{21}$$