# Empirical and Definitional Content of Newton's Laws

I'd like to get a better insight into which aspects of Newton's laws are definitional and which are falsifiable. And moreover, of the definitional aspects, why these are good definitions.

Netwon's laws can be phrased as follows (from Wikipedia):

First law: In an inertial frame of reference, an object either remains at rest or continues to move at a constant velocity, unless acted upon by a force.

Second law: In an inertial frame of reference, the vector sum of the forces F on an object is equal to the mass m of that object multiplied by the acceleration a of the object: F = ma.

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.

In the first law, neither force nor frame of reference have been defined. In the second law, force is given a relationship to mass and acceleration. We might presume that acceleration can be measured and needs no definition. However, mass and force are defined through this equation as far as I understand. Therefore, the first and second law can't yet make any empirical predictions, as we still have two unknown quantities for each measurement of acceleration. In other words, the first two laws (along with the definition of an inertial frame) define mass and force. The third law implicitly tells us that bodies can apply forces to each other (which wasn't apparent from the first two laws), but more explicitly it adds a constraint of reciprocity between the forces of interacting bodies. This still doesn't seem to provide any empirical content given that force and mass are underdetermined from measurements of acceleration according to these laws. However, once we finally posit a functional form for force, for example via Netwon's law of gravitation, it looks as though we have something empirically testable - that is, using measurements of acceleration, we can falsify the theory. So would it be correct to state that Newton's 3 laws are purely definitional, and that only with the additional specification of further laws defining the forces do they become testable? If so, can we claim that this choice of definitions is a good one, as opposed to some other choice?

Mentor
So would it be correct to state that Newton's 3 laws are purely definitional, and that only with the additional specification of further laws defining the forces do they become testable?
There is a considerable amount of differing opinions on the topic in the literature, so I think you have some flexibility in your preference. I have never seen anyone go as far as to say that all three together are definitional with no experimental content.

My personal preference is to consider the first law as defining an inertial frame, and the second law as defining a force. Then the third law is experimentally testable and contains the physical content of Newton’s laws

There is a considerable amount of differing opinions on the topic in the literature, so I think you have some flexibility in your preference. I have never seen anyone go as far as to say that all three together are definitional with no experimental content.

My personal preference is to consider the first law as defining an inertial frame, and the second law as defining a force. Then the third law is experimentally testable and contains the physical content of Newton’s laws

As far as I can see, the third law only becomes empirically testable once a force law has been introduced (e.g., Netwon's law of gravitation). Even then, Newton's law of gravitation appears to entail Newton's third law by symmetry. Before a force law has been introduced, it seems that Newton's third law acts more as a constraint on the possible kinds of force laws that one could postulate.

However, mass and force are defined through this equation as far as I understand.
That's a matter of interpretation. One could also assume force to be already defined, for example in terms of Hooke's Law.

Mentor
As far as I can see, the third law only becomes empirically testable once a force law has been introduced (e.g., Netwon's law of gravitation).
I have never seen any reference that makes this claim. Have you?

That's a matter of interpretation. One could also assume force to be already defined, for example in terms of Hooke's Law.

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. Yet 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 situations using the definitions in Netwon's first two laws. Newton's third law tells us that we should only look for theories of forces which obey a certain symmetry.

We could instead start with Hooke's law, then arrive at a concept of mass and intertial frame using Newton's first two laws. Perhaps by comparing objects of different masses we might even arrive at a uniquely specifying account of the mass of each object by that definition. Then we could assume that Hooke's law is specific only to one particular scenario, but that Newton's 3 laws are more general, and start to search for theories of forces in other scenarios.

To my mind the two approaches look like the same thing.

I have never seen any reference that makes this claim. Have you?

No, but it seems like something which can be reasoned directly from the laws themselves. The first two laws tell us that F=ma. If we assume that we can only measure a (or even also position and velocity), then Newton's third law doesn't make any testable predictions. Only when we write down an equation for the dependence of F on position (and other variables) does something testable emerge.

Mentor
No, but it seems like something which can be reasoned directly from the laws themselves.
That isn’t the standard here on PF. We require that all posts be consistent with the professional scientific literature. Your personal reasoning is not acceptable if it is inconsistent with the literature.

Here is an example I like.

https://www.worldscientific.com/doi/pdf/10.1142/9789810248154_0001

Calkin does not seem to share your opinion.

That isn’t the standard here on PF. We require that all posts be consistent with the professional scientific literature. Your personal reasoning is not acceptable if it is inconsistent with the literature.

Fair enough, no need to get your knickers in a twist!

Here is an example I like.

https://www.worldscientific.com/doi/pdf/10.1142/9789810248154_0001

Calkin does not seem to share your opinion.

If I understand correctly, what your reference tells us is that for an isolated pair of particles the ratio of accelerations should be constant according to Newton's third law. Indeed that would imply some empirical content in the 3rd law.

Mentor
for an isolated pair of particles the ratio of accelerations should be constant according to Newton's third law
Yes, and regardless of the nature of the interaction.

DrStupid
In the first law, neither force nor frame of reference have been defined.

For mass and force Newton provided definitions. But they are not very useful.

He introduced frames of references in a different chapter prior to the laws of motion. It is quite long-winded and cannot be summarized in a handy definition. It is mainly about an absolute frame of reference (defined by the fixed stars) and moving frames of reference that cannot be distinguished from the absolute frame by experimental observations. This is quite similar to inertial frams of reference as we know them today. If you read the laws of motion without this background, inertial frames are indeed not yet defined with the first two laws.

In other words, the first two laws (along with the definition of an inertial frame) define mass and force.

The first two laws (in the wording you mentioned them above) are not sufficient for an implicit definition of mass, force or inertial frames. The third law and some additional restrictions (e.g. principle of relativity and isotropy) are required as well.

The third law implicitly tells us that bodies can apply forces to each other (which wasn't apparent from the first two laws), but more explicitly it adds a constraint of reciprocity between the forces of interacting bodies. This still doesn't seem to provide any empirical content given that force and mass are underdetermined from measurements of acceleration according to these laws.

Newton supported the 3rd law with experimental observations showing that momentum is conserved. This is empirical content.

However, once we finally posit a functional form for force, for example via Netwon's law of gravitation

The law of gravitation is based on the definition of force. It is not part of the definition.

So would it be correct to state that Newton's 3 laws are purely definitional, and that only with the additional specification of further laws defining the forces do they become testable?

It is definitional in the sense that it distinguishes forces from fictious forces. But conservation of momentum is not definitional. That can be tested by experiments.

Yes, and regardless of the nature of the interaction.

For mass and force Newton provided definitions. But they are not very useful.

He introduced frames of references in a different chapter prior to the laws of motion. It is quite long-winded and cannot be summarized in a handy definition. It is mainly about an absolute frame of reference (defined by the fixed stars) and moving frames of reference that cannot be distinguished from the absolute frame by experimental observations. This is quite similar to inertial frams of reference as we know them today. If you read the laws of motion without this background, inertial frames are indeed not yet defined with the first two laws.

The first two laws (in the wording you mentioned them above) are not sufficient for an implicit definition of mass, force or inertial frames. The third law and some additional restrictions (e.g. principle of relativity and isotropy) are required as well.

Newton supported the 3rd law with experimental observations showing that momentum is conserved. This is empirical content.

The law of gravitation is based on the definition of force. It is not part of the definition.

It is definitional in the sense that it distinguishes forces from fictious forces. But conservation of momentum is not definitional. That can be tested by experiments.

In response to both posts, I have one final question. It seems that Newton's third law only has testable content if we can independently verify that we are in an intertial frame of reference. Ratios of acceleration are not constant in a non-inertial frame, and nor is momentum conserved. Newton's three laws (in the form that I originally posted them) don't seem to provide a way to test whether we are in an intertial frame other than by checking if momentum is conserved. That would make the three laws purely definitional. On the other hand DrStupid mentioned an independent definition of intertial frames relative to the "fixed stars". If we allow for that, then we would have testable predictions from Newton's third law. Is this correct?

DrStupid
Newton's three laws (in the form that I originally posted them) don't seem to provide a way to test whether we are in an intertial frame other than by checking if momentum is conserved. That would make the three laws purely definitional.

Yes, you can define inertial frames this way. But you need to add that the frames of reference must be global because otherwise locally free falling frames of reference would be inertial too. That wouldn't fit well to the law of gravitation which says that gravity is an interactive force. With the definition of a global absolute frame of reference using the fixed stars Newton avoided this problem.

Mentor
Newton's three laws (in the form that I originally posted them) don't seem to provide a way to test whether we are in an intertial frame
The first law is often understood to do exactly that.

The first law is often understood to do exactly that.

The first law would do that if we independently knew what the forces acting on a body were, but if both force and inertial frame are to be defined through Newton's 3 laws then it doesn't seem there are any testable predictions. Taken in isolation, Newton's first law only defines a relationship between inertial frame and force - using the 1st law alone, given an accelerating object we can't distinguish between the case where we are in a non-inertial frame or the case in which we are in an inertial frame with a force acting on the object.

DrStupid
Taken in isolation, Newton's first law only defines a relationship between inertial frame and force - using the 1st law alone, given an accelerating object we can't distinguish between the case where we are in a non-inertial frame or the case in which we are in an inertial frame with a force acting on the object.

I agree with you, but after a long discussion about this topic I got the impression that the idea that you need the 3rd law to identify inertial frames is not very popular in this forum. However, I didn't got a clear answer how to do it instead.

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I think the logic in a modern formulation is as follows, and I hope, we won't have a big battle about "reference frames" again.

Everything starts with a theory about space and time. In the case of Newtonian physics you have the assumption of "absolute time" and "absolute space". Time is just an oriented one-dimensional differentiable manifold, i.e., after introducing a chart, isomorphic to the real numbers, and absolute space is a Euclidean affine manifold. "Absolute" means that this structure is "rigid" in the sense that no physical process can ever change this spacetime structure.

Then Lex I says that by assumption there exists an inertial frame, and from this of course it follows that there are arbitrary many inertial frames all connected by Galilei transformations from one to another. An inertial frame is thereby defined by Lex I, i.e., that a free body (i.e., a body upon which no forces act) moves rectilinear and uniformly, i.e., with constant velocity.

It was Lange who put this into a non-circular way by showing that you need three free bodies with trajectories moving through a common point which are not along the same line to operationally define an inertial reference frame. To check that it really is one you have to verify that any fouth free body in any direction moves with constant velocity against the so established reference frame.

Lex II then quantifies forces and inertia in the usual way

Lex III is from a modern point of view following from the symmetries of the Galilei-Newtonian spacetime manifold, i.e., from homogeneity of absolute space leading to the necessity that the total momentum of a closed system must be conserved (following from Noether's theorems). In this context the absoluteness of time is also important, because it implies the typically Newtonian idea of "action at a distance". Only then Lex III can be valid for a closed set of point particles alone.

This becomes already clear when you think about Special Relativity theory (SRT). You can get to the idea of SRT by asking how a spacetime model must look like, where you do not necessarily have absolute time and absolute space but only assume the validity of Lex I in the above sense, i.e., the existence of an inertial reference frame and thus arbitrary many moving with constant velocity against each other and otherwise keep the space for any inertial observer as an affine Euclidean manifold. Then the analysis about how to transform from one inertial frame to another leads to either the Galilei transformations or the Poincare transformations. The symmetry group then let's you construct the mathematical structure of the spacetime model. In the case of Galilei transformations you get the fiber-bundel structure of absolute space and time, i.e., Galilei-Newton spacetime and for the Poincare group you get Einstein-Minkowski spacetime of SRT.

It is then an empirical question, which spacetime model fits better with the observations, and it's well known that Einstein-Minkowski spacetime "wins" particularly with respect of electromagnetic phenomena, and that's why Einstein was led to SRT by analyzing the quibble with the lack of Galilei invariance of Maxwell's electromagnetic theory as well as the many null results concerning the existence of an aether (Trouton Noble, Michelson-Morley,...).

Also there is some tension between SRT and Newton's Lex III. The resolution of this tension is that the 10 conservation laws from the spacetime symmetries still hold in SRT as in Newtonian physics but the interactions are localized through the mathematical description of Faraday's important idea of local laws in terms of fields (like in his case the electromagnetic field), i.e., the field is a fundamental entity of nature as is matter and it's a dynamical entity too. Today, the most successful theories are indeed local field-theoretical equations obeying the spacetime structure given by Einstein-Minkowski spacetime, of course in quantized form, i.e., in terms of local relativistic quantum field theory.

The only thing, as far as we know today, that models cannot describe is the gravitational interaction, which needs of course General Relativity.

Mentor
The first law would do that if we independently knew what the forces acting on a body were
All that is required is to be able to identify non-interacting bodies. If you can independently know what forces are acting on a body then you can use the motion of interacting bodies to predict what the motion of non-interacting bodies would be.

vanhees71
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Well, but even Newton himself was aware of this somewhat "circular" definitions arising from his postulates. Already at his time there was a heavy debate about the question how an inertial frame can be operationally determined. Indeed, the problem with absolute space and absolute time is how to observe them, if apparently one can determine the motion of one body only relative to other bodies. Newton's answer is the famous bucket argument, i.e., that a rotating bucket filled with water shows a parabolic rather than a plane surface due to what we'd call inertial forces today in the rotating frame. On the other hand the question is, "rotating relative to what". And of course in this case the bucket was simply rotating relative to the Earth, which is to some approximation indeed an inertial frame (when taking of course the gravitational force of the Earth on the objects described relative to it into account, and indeed the parabolic shape of the corotating fluid in the bucket is a result of the inertial (centrifugal) force and the gravitational force). Of course on the other hand you can demonstrate, e.g., by the Foucault pendulum experiment that the Earth-fixed frame is of course not really an inertial frame.

At the end of course you always have somehow to operationally define an inertial frame (and thus in fact an entire equivlence class of inertial frames).

All that is required is to be able to identify non-interacting bodies.

If two non-interacting bodies were accelerating at a constant rate, how would you know whether they were acclerating due to a common force (e.g., gravity) or due to a non-inertial frame of reference?

If you can independently know what forces are acting on a body then you can use the motion of interacting bodies to predict what the motion of non-interacting bodies would be.

I'm not sure if I understand you here. Could you spell it out more clearly?

@vanhees71 - Thanks for the informative post, there's a lot for me to unpack there. Are there any references for this kind of approach? I'm still not clear on which parts are simply ironing out mathematical details to provide non-circular definitions versus those parts that make testable empirical predictions.

If two non-interacting bodies were accelerating at a constant rate, how would you know whether they were acclerating due to a common force (e.g., gravity) or due to a non-inertial frame of reference?
If they are affected by Newtonian Gravity, then they aren't non-interacting, because Newtonian Gravity is an interaction force.

If they are affected by Newtonian Gravity, then they aren't non-interacting, because Newtonian Gravity is an interaction force.

I had taken non-interacting to mean not interacting with each other. Is the intended meaning that no force is acting on either of the bodies?

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.
Yes, just like you have definitions for specific types of energy: kinetic, potential etc.
- To test the Law of Energy Conservation you use specific energy definitions.
- To test Newton's Laws you use specific force definitions.

etotheipi
Mentor
If two non-interacting bodies were accelerating at a constant rate, how would you know whether they were acclerating due to a common force (e.g., gravity) or due to a non-inertial frame of reference?
In this approach (the one I linked to above) the first law says that non-interacting bodies do not accelerate. In standard Newtonian physics gravity is an interaction so a non-interacting body cannot be accelerating due to gravity. In Newton-Cartan theory gravity is geometrized so a non-interacting body can be subject to gravity and is in free fall.

I'm not sure if I understand you here. Could you spell it out more clearly?
Suppose that you are on Earth and using standard Newtonian physics where gravity is an interaction. You know that gravitational acceleration is 9.8 m/s^2 downward. So you can drop an uncharged and non-magnetic object and define an inertial frame as one where it accelerates downward at 9.8 m/s^2. It is not a non-interacting object in this situation, but since you know the force on it you can determine what a true standard Newtonian non-interacting object would do.

etotheipi
Yes, just like you have definitions for specific types of energy: kinetic, potential etc.
- To test the Law of Energy Conservation you use specific energy definitions.
- To test Newton's Laws you use specific force definitions.

Surely that would constitute a test of the conjunction of Newton's laws and those specific force definitions. A falsification would then only falsify Newton's laws OR those specific force definitions.

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On the textbook level the best discussion, though pretty short, is still found in A. Sommerfeld, Lectures on Theoretical Physics, Vol. 1 (Mechanics). There's also a nice discussion in Einstein's two textbooks on special and general relativity.

etotheipi
DrStupid
On the other hand the question is, "rotating relative to what".

Relative to the inertial frame or something that is at rest in an inertial frame.

And of course in this case the bucket was simply rotating relative to the Earth, which is to some approximation indeed an inertial frame (when taking of course the gravitational force of the Earth on the objects described relative to it into account, and indeed the parabolic shape of the corotating fluid in the bucket is a result of the inertial (centrifugal) force and the gravitational force). Of course on the other hand you can demonstrate, e.g., by the Foucault pendulum experiment that the Earth-fixed frame is of course not really an inertial frame.

Newton was aware of it and it doesn't limit the validity of the bucket argument. On very large scales the rotation turns a spherical surface into an ellipsoid. Newton even calculated the oblateness of the Earth due to its rotation for the case of homogeneous density.

Surely that would constitute a test of the conjunction of Newton's laws and those specific force definitions. A falsification would then only falsify Newton's laws OR those specific force definitions.
You cannot falsify definitions.

DrStupid
You cannot falsify definitions.

As long as they are not self-contradicting.

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2022 Award
Relative to the inertial frame or something that is at rest in an inertial frame.

Newton was aware of it and it doesn't limit the validity of the bucket argument. On very large scales the rotation turns a spherical surface into an ellipsoid. Newton even calculated the oblateness of the Earth due to its rotation for the case of homogeneous density.
I said that Newton was well aware of all this and also about the circularity of his postulates. The point is that at the end you have to operationally define a reference frame that is (to a sufficient accuracy) inertial. You can only calculate the ellipsoidal shape of the idealized Earth due to its rotation by assuming (!) that it rotates relative to some inertial frame. Here a very good approximation for this inertial reference frame is the reference frame defined by the rest frame of the fixed stars.

etotheipi
DrStupid
I said that Newton was well aware of all this and also about the circularity of his postulates.

I do not see the circularity.

You can only calculate the ellipsoidal shape of the idealized Earth due to its rotation by assuming (!) that it rotates relative to some inertial frame.

Why do I need to assume that? I can also calculate the shape assuming that the Earth doesn't rotate. The comparision of the result with experimental observations will show that my assumption was wrong and that the rest frame of Earth is in fact rotating.

You cannot falsify definitions.

You seem to have suggested that Newton's law of gravitation, along with Hooke's law etc., are definitions. It seems more plausible to me that Newton's 3 laws are the definitions, in which case these specific force laws are not definitions. We may have some freedom to choose which parts of the theory are definitions and which aren't, but in the end we need to put down a sufficient number of constraints in order to arrive at anything testable.

In this approach (the one I linked to above) the first law says that non-interacting bodies do not accelerate. In standard Newtonian physics gravity is an interaction so a non-interacting body cannot be accelerating due to gravity.

It says they do not accelerate in an inertial frame. But we still have to figure out whether we are in such a frame. I don't see how we can identify these non-interacting bodies until we decide whether we are in an inertial frame of reference.

Suppose that you are on Earth and using standard Newtonian physics where gravity is an interaction. You know that gravitational acceleration is 9.8 m/s^2 downward. So you can drop an uncharged and non-magnetic object and define an inertial frame as one where it accelerates downward at 9.8 m/s^2. It is not a non-interacting object in this situation, but since you know the force on it you can determine what a true standard Newtonian non-interacting object would do.

Sure, we have agreed already that once you already have knowledge of the external force you can resolve these things. But that isn't given by Newton's laws.

DrStupid
It seems more plausible to me that Newton's 3 laws are the definitions, in which case these specific force laws are not definitions.

I would even say that the law of gravitation is a prime example of a law of nature - a generalisation of experimetal observations.