# Empirical and Definitional Content of Newton's Laws

In summary: Sure, there are laws which define force in specific scenarios, such as Hooke's law, Newton's law of gravitation, or Maxwell's laws of electromagnetism.However, Newton's laws are taken to be true for all forces. In this sense they feel like a framework for defining concepts of mass and force - the mass of an object is assumed to be the same across all kinds of force, in which case the goal is to find theories which describe forces in various...
avicenna said:
...m₁ too has relative acceleration a₂ relative to m₂ ...
No, the acceleration in the 2nd law is not relative to a second mass. The 1st and 2nd laws do not even mention a second mass, only the 3rd does. So it doesn't make any sense that the 3rd follows from the 1st and 2nd.

Dale
A.T. said:
No, the acceleration in the 2nd law is not relative to a second mass. The 1st and 2nd laws do not even mention a second mass, only the 3rd does. So it doesn't make any sense that the 3rd follows from the 1st and 2nd.
The OP used this form of the 3rd law, action and reaction between two bodies.
Third law: When one body exerts a force on a second body, the second body simultaneously exerts a force equal in magnitude and opposite in direction on the first body.

The Principia original may be: "For every action, there is an equal and opposite reaction"

Let's consider m₁ acts with the force f on m₂. Assume the inertial reference frame with m₁ as origin. From the 2nd law, m₂ has an acceleration a₂ = f/m₂ in this frame.

Now we take all motion relative to frame (inertial?) with origin at m₂. In this frame, m₁ has an acceleration which magnitude is the same a₂= f/m₂. If this frame of m₂ is inertial, then there is a force of magnitude f acting on m₁.

avicenna said:
Assume the inertial reference frame with m₁ as origin...
The rest frame of m1 is not inertial, if there is a force acting on m1.

avicenna said:
...If this frame of m₂ is inertial,...
See above.

A.T. said:
The rest frame of m1 is not inertial, if there is a force acting on m1.See above.
I think I am wrong. The 3rd law is another necessary axiom. It definitely is not as a testable law. It is the 3rd axiom that allows Newton's gravitational law to act mutually between two bodies.

avicenna said:
It definitely is not as a testable law.
It is testable:

Start with two bodies interacting with each other but otherwise non-interacting. Measure their momentum over time wrt some inertial frame. If their total momentum changes then the third law is experimentally falsified.

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vanhees71
Dale said:
It is testable:

Start with two bodies interacting with each other but otherwise non-interacting. Measure their momentum over time wrt some inertial frame. If their total momentum changes then the third law is experimentally falsified.
Your experiment cannot be said to be a test of the third axiom alone, but rather a test of Newton's three axioms together; i.e. Newton's three laws of motion holds when there is no change of momentum over time.

The reasoning is this; the law of conservation of momentum is a consequence of applying the three laws together. If anyone of the three axioms cannot be satisfied or fails, then there would be a change in momentum over time; .e.g. if the 2nd law fails, then force will not be proportional to rate of change of momentum; this would also imply f ≠ ma and your experiment would fail to verify the three laws of motion.

weirdoguy
avicenna said:
Your experiment cannot be said to be a test of the third axiom alone, but rather a test of Newton's three axioms together
That is a world of difference from your blanket statement that it is definitely not testable.

I frankly don’t care if you consider the laws separately or together. Either way, you definitely do end up with something that is testable.

And in any case it certainly “can be said” to be a test of Newton’s 3rd law alone by considering the first two laws to be definitions and therefore not testable. Perhaps that is not your preferred formulation but “cannot be said” is vastly overstating the case.

Dale said:
That is a world of difference from your blanket statement that it is definitely not testable.

I frankly don’t care if you consider the laws separately or together. Either way, you definitely do end up with something that is testable.

And in any case it certainly “can be said” to be a test of Newton’s 3rd law alone by considering the first two laws to be definitions and therefore not testable. Perhaps that is not your preferred formulation but “cannot be said” is vastly overstating the case.
I think there is good reason why some would translate the Latin "Lex" as axiom and not law. The conventional use of the word law in physics is a standalone testable law as in Coulomb's law, Ohm's law.

Newton's three laws of motion are closer to axioms in mathematics; they are the basic framework wherein calculations may be made in mechanics. There are no priority in the three axioms; they are the set of axioms. If the momentum experiment fail, there is no way to determine which of the axioms failed; it could be the 3rd axiom is ok, but the 2nd axiom failed.

So I would prefer to view Newton's three laws together and as the starting foundation of mechanics.

avicenna said:
The conventional use of the word law in physics is a standalone testable law as in Coulomb's law, Ohm's law.
In that sense Newton’s 3rd law is indeed a law. In Ohm’s law current and voltage are defined elsewhere. Similarly in Newton’s 3rd law inertial frames and forces are defined elsewhere. The law (Ohm’s or Newton’s 3rd) is then testable. If you allow Ohm’s law as a law then you have little basis to exclude Newton’s 3rd law as a law.

avicenna said:
I think there is good reason why some would translate the Latin "Lex" as axiom and not law. ...

Newton's three laws of motion are closer to axioms in mathematics; they are the basic framework wherein calculations may be made in mechanics. There are no priority in the three axioms; they are the set of axioms. If the momentum experiment fail, there is no way to determine which of the axioms failed; it could be the 3rd axiom is ok, but the 2nd axiom failed.
All of this, at best, is a matter of personal preference, and it is a preference that I do not share. Particularly the claim that “Newton's three laws of motion are closer to axioms in mathematics“. Since at a minimum together they are experimentally testable, such a claim seems problematic at best.

I have not even seen a professional scientific reference which makes this claim. Do you have such a reference or is this pure personal speculation?

avicenna said:
So I would prefer to view Newton's three laws together and as the starting foundation of mechanics.
So say that. Your blanket claims like “it cannot be said ...” are simply wrong.

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There is no empirical content. Newton's laws are axioms, that not goes from reality, like axioms of mathematics or church dogmatic or something like that

weirdoguy
You have axioms in mathematics. There are no axioms in physics, just some fundamental laws which are based on observations.

etotheipi
vanhees71 said:
You have axioms in mathematics. There are no axioms in physics, just some fundamental laws which are based on observations.

It's not true for modern physics at all and "classical" physics in particular. Where do you "observe" Newton's laws? In other words, how you deduce it from reality concretely?

weirdoguy
fightingphysics said:
Where do you "observe" Newton's laws?

In the lab during second semester of first year of studying physics at Warsaw University

Dale, etotheipi and vanhees71

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