Is Newton's third law a consequence of first and second law?

Question

Refer to the diagram shown above

Question

Assume a system of two objects ($1$ and $2$) which are applying force on one another ($F_{12}$ and $F_{21}$), which can be zero, (read $F_{12}$ as force on object $1$ by $2$).

Now by Newton's Second law the rate of change of momentum of Object 1 is given by:

$$\frac {d \mathbf p_1}{dt}=\mathbf F_{12}+\mathbf F_1^{ext} \tag 1$$

Similarly for Object 2 is:

$$\frac {d \mathbf p_2}{dt}=\mathbf F_{21}+\mathbf F_2^{ext} \tag 2$$

Adding equation $(1)$ and $(2)$ we get:

$$\frac {d (\mathbf p_1+ \mathbf p_2)}{dt}=\frac {d \mathbf P}{dt}= (\mathbf F_{12} + \mathbf F_{21})+(\mathbf F_1^{ext}+\mathbf F_2^{ext})$$

Now since Newtons First Law says that unless an external unbalanced force acts on a system it continues its initial state of rest or uniform motion. This means that in the absence of external force the momentum of a system is constant. This means that $$\mathbf F_1^{ext}+\mathbf F_2^{ext}=0$$ $$\Rightarrow \frac {d \mathbf P}{dt}=0$$ $$\mathbf F_{12} + \mathbf F_{21}=0$$ i.e., $$\mathbf F_{12}=- \mathbf F_{21}$$

Now this clearly means that Newton's Laws are self consistent but:

• Why doesn't it mean that Newton's third law is derived from the first and second law?

• In general (as shown above) why doesn't the consistency of one law with other does not imply the derivability of the latter from former?

Answer 1

As pointed out in the comments, your argument cannot be applied to systems with more than 2 interacting bodies. For $N$ bodies, you will arrive at $$\sum_i^N\sum_{j\neq i}^N\mathbf F_{ij}=0$$ which does not specify $\mathbf F_{ij}=-\mathbf F_{ji}$ for all $i,j$ for $N>2$.

Therefore, Newton's third law is not obtained here. It must be an additional assumption.

Answer 2

I would like to rephrase this question into the following form:

Is three laws the minium, or is a system of two axioms also possible?
I think the answer to that is yes.

Of course, implicitly you are already using the axioms in the form I describe below, I'm only making it explicit.

First let me propose a form of the first law that is similar in form to Kepler's law of areas.

Axiom:
Motion of a particle in the absence of a net force:
in equal intervals of time the motion will cover equal intervals of spatial distance.

The above axiom is stretching to deal with multiple things, among them properties of space and properties of time.

Since the second law (F=ma) will cover any change of motion (including absence of change), the first law doesn't have to make a statement about motion. Instead, the first law can be both narrowed down and expanded to deal exclusively and explicitly with space and time.

Axiom 1:
Physical space has the same uniformity and the same symmetries as 3-dimensional Euclidean space.
Physical time has the same uniformity and the same symmetry as 1-dimensional Euclidean space.

Axiom 2:
F = ma

It seems to me that in the above form the axioms are sufficient to imply newton's third law.

Thought demonstration:
You are in a spacecraft, floating in space, and you want to push another spacecraft away. If the other spacecraft has much more inertial mass than your own, then you pushing will result mostly in you pushing yourself away from the common center of mass.

To do any pushing around you need leverage, and in space the only leverage you have is your own inertial mass. For the second law it makes no difference who is doing the pushing. Since symmetry of space has been asserted in the first axiom the third law follows logically.

The above form of the axioms also has the following advantage: when transitioning to relativistic physics the replacement has exactly the same structure, only different content.

Axiom 1:
Physical spacetime has the same uniformity and the same symmetries as mathematical Minkowski spacetime.

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LATER EDIT

Expanding the above thougth demonstration:

A team of programmers is commissioned to program a simulation of, say, the motions of a solar system.

Assume the members of the team have no prior knowledge of theory of motion.
The team is given the following pieces of information:
- The first and second law of motion
- Newton's universal law of gravity

The team is provided with the information that in physics it is invariably seen that if there is a formula relating three quantities all three permutations of the formula are causally valid.

More generally, in all of physics there is no known case of a one-way-door.

Example of every-direction-is-causally-valid: Ohm's law: V=IR
Whenever you know any two of those of the quantities you apply that formula to obtain the third.

The team members discuss the expected outcome of the following case: two spacecrafts, suspended in space. A cable extends between the spacecrafts. One spacecraft starts reeling in the cable. What will happen?

The team members assess as follows:
The reeling-in spacecraft is pulling itself towards the common center of mass of the two spacecrafts. The reason the reeling-in spacecraft can exert a force at all is because of its own inertial mass. The larger the inertial mass of the reeling-in spacecraft, the smaller its own acceleration towards the common center of mass. It follows that distinction between the-reeling-in-spacecraft and spacecraft-being-reeled-in is moot. The result will be identical so you can't tell the difference anyway. Taking it from there they have all they need to program a correctly running simulation.

Let the inertial mass of one spacecraft be 'm1' and the other 'm2'

The team members infer: m1*a1 = -m2*a2

In physics, when one does assessments one must always keep an awareness of physics background knowledge. All of that should infuse any assessments under hand.

And yeah, back in the days of Newton and his contemporaries there was much less background knowledge available. So in that sense in Newton's time the case for adding the third law was stronger than it is now.