Empirical and Definitional Content of Newton's Laws

[DrStupid]

I don't see how we can identify these non-interacting bodies until we decide whether we are in an inertial frame of reference.

In theory it would be clear if there is just a single body. In practice there is always the possibility of unknown interactions. But we try to describe nature as simple as possible. That includes not assuming hidden interactions as long as we can avoid it.

 

[DrStupid]

I believe Newton called the 3 laws " Axioms"

I would translate "Lex" into law but I'm not an expert in Latin.

 

[A.T.]

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.

Yes, and post #23 describes how Newton's Laws are testable in analogy to Energy Conservation.

 

[madness]

Yes, and post #23 describes how Newton's Laws are testable in analogy to Energy Conservation.

I'm not sure an analogy to Energy Conservation makes the case.

The conjunction of Newton's laws and a choice of force law may be testable. But I'm not sure whether one could maintain testability without first specifying all 3 laws and the force law.

 

[hutchphd]

I would translate "Lex" into law but I'm not an expert in Latin.

I was working from the chapter title Axiomata sive leges Motus where I think the translation holds good. But he does then enumerate the "laws". Alas I am devoid of any formal training in Latin.

 

[Dale]

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.

As I said before, this approach assumes that you can already identify non-interacting bodies. Then you simply look at their motion. If it is uniform motion then the frame is inertial per the 1st law. I.e. assuming you can identify non-interacting bodies then the 1st law defines inertial frames.

If you cannot identify non-interacting bodies then it will not work. For this reason the Newton-Cartan approach is nice because non-interacting bodies are easy to identify simply using an accelerometer, similar to the approach of GR.

 

[A.T.]

I'm not sure an analogy to Energy Conservation makes the case.

The conjunction of Newton's laws and a choice of force law may be testable.

Energy Conservation is also only testable, if you define how to calculate specific forms of energy.

 

[madness]

As I said before, this approach assumes that you can already identify non-interacting bodies. Then you simply look at their motion. If it is uniform motion then the frame is inertial per the 1st law.

If you define non-interacting as being subject to no external force, then it seems fairly trivial that one could check whether they were in an intertial frame by measuring the acceleration of a known non-interacting body. But I don't find that very helpful or insightful because checking whether a body is non-interacting or checking whether we are using a non-inertial frame appear equally impossible on the basis of Newton's laws, at least before we define a functional form for the forces in the system.

 

[madness]

Energy Conservation is also only testable, if you define how to calculate specific forms of energy.

Sure, the conjunction of the statements "energy = X" and "energy is conserved" is testable, as is the conjunction of Newton's laws and the specific laws for force.

 

[A.T.]

Sure, the conjunction of the statements "energy = X" and "energy is conserved" is testable, as is the conjunction of Newton's laws and the specific laws for force.

And if you just call them all "statements", like you did for energy, then you don't have to worry which are "laws" and "definitions".

 

[madness]

And if you just call them all "statements", like you did for energy, then you don't have to worry which are "laws" and "definitions".

Perhaps, except that some of the "statements" are supposed apply to all possible forces and scenarios whereas others only in particular scenarios. In any case, my main concern was whether we really need all of the statements to get to something empirically testable. As far as I have understood, some posters have argued that Newton's 3 laws already get us to something testable, but it seems that this is only true if you also have one of the following extra facts: 1) Knowledge of which frames are intertial 2) Knowledge of some particles that are "non-interacting" 3) A definition for a specific force.

 

[DrStupid]

Energy Conservation is also only testable, if you define how to calculate specific forms of energy.

But in contrast to energy there is only one form of momentum with a very simple definition. That makes it quite easy to test conservation of momentum without any specific force law.

 

[vanhees71]

I do not see the circularity.
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.

The circularity is that you need on the one hand an inertial frame which is operationally defined by the Lex I, for which you need the notion of a free particle, i.e., a particle which is not subject to the action of forces. To define forces you need Lex II which uses the definition of an inertial frame.

The resolution, also in view of the hitherto most comprehensive spacetime model, which is General Relativity, in my opinion is that you must start with a postulate on the spacetime model and then use it to find operational realizations of inertial reference frames. In Newtonian physics you start with absolute space and time, which however cannot be operationally defined, and indeed there's no way to distinguish different inertial frames, which is due to the fact that the symmetry group of Galilei-Newton spacetime is the 10D Galilei group of transformations. Having identified the symmetry group of the spacetime model you can reconstruct this model (this line of thought you can trace back to Riemann and Klein's Erlanger program in the mathematical foundation of geometry (or rather different kinds of gemoetries), which was worked out famously later by Noether in her famous 1918 paper on symmetries and conservation laws).

The next step in the development was famously Einstein's solution of the problem with the lack of Galilei invariance of Maxwell electrodynamics on the one hand and the failure to establish a preferred reference frame concerning electromagnetic phenomena on the other hand. Many physicists have been thinking before that this is the long-sought possibility for an operational approach to define Newton's absolute space and time as a kind of rest frame of the conjectured aether. This was disproven around this time (around 1900) by various experiments, including the most famous Michelson-Morley experiment but also the Trouton-Noble experiment.

Einstein just reinterpreted all these failed attempts by just assuming the full validity of the special principle of relativity but making it compatible with Maxwell electrodynamics. He found out that all you need to assume in addition is that the speed of electromagnetic waves is independent of the motion of the source of the waves relative to any inertial reference frame, and of course he used this to derive the Lorentz transformations and this finally lead Minkowski to construct the corresponding spacetime model, the Minkowski space as an affine pseudo-Euclidean 4D spacetime.

The last (yet) necessary adoption of the spacetime model then was Einstein's attempt to incorporate gravity into the relativistic picture lead him to the discovery of General Relativity in using the (weak and strong) equivalence principle. This lead to the reinterpretation of the gravitational interaction in terms of a pseudo-Riemannian manifold, where the inertial frames are only definable locally with the general covariance as a gauge symmetry, as we'd interpret it today in view of our experiences with gauge theories in the modern sense as a mathematical tool to localize global symmetries. In this case what's "localized" is the Lorentz group, and as long as one considers only the macroscopic physics of classical relativistic (continuum) mechanics and electrodynamics you are able to reconstruct Einstein's general relativistic spacetime model as a Lorentzian (pseudo-Riemannian manifold with a metric of signature (1,3) or equivalently (3,1)).

The operational definition of a local inertial reference frame from the point of view of GR is a freely falling body with a non-rotating tetrad (Fermi-Walker transported) defining this frame. The extension of this local inertial frame is determined by the scales over which tidal gravitational forces can be neglected. A nice example of such a local inertial reference frame is the International Space Station, which as far as I know is the best place to make microgravitation experiments, i.e., the best operationally defined local inertial reference frame we have today.

 

[DrStupid]

The circularity is that you need on the one hand an inertial frame which is operationally defined by the Lex I, for which you need the notion of a free particle, i.e., a particle which is not subject to the action of forces. To define forces you need Lex II which uses the definition of an inertial frame.

I still don't see a circularity but an axiomatic definition instead. The laws of motion alone might not be sufficient to distinguish rotating frames from non-rotation frames but there are other restrictions like isotropy that would be violated in the rotating frames. If you manage to invent interactive forces to describe rotating systems they would define a preferred direction.

 

[Dale]

I don't find that very helpful or insightful

That is fine. There are many approaches in the literature. Pick one that works for you. I like this one, but I also have seen others I like.

You may be interested in the Newton-Cartan approach where gravity is not an interaction so identifying non interacting bodies is a simple matter of attaching an accelerometer.

some posters have argued that Newton's 3 laws already get us to something testable

Note that, in my case it isn’t “some posters” arguing, it comes from the literature. There is a lot of variety on this topic, which is an indication that it is mainly just a personal preference. So just find one that you like.

 

[vanhees71]

I think the important point is to realize that you need a spacetime model in order to be able to measure locations and times of events. On the other hand this spacetime model only makes physically sense when you are able to also operationally define the quantities appearing in the mathematical spacetime model. Then you can check the assumptions made in assuming the specific structure of the spacetime model for consistency.

History shows that indeed spacetime models can be empirically tested: In 1905 it became clear that the so far used Newtonian spacetime model has its limited range of applicability, i.e., it is an approximation for a more accurate spacetime model, Minkowski space, which itself is again an approximation since to describe gravity the within relativistic physics needed another spacetime model, and again it can be tested empirically, nowadays partially with very high accuracy (Pulsar Timing, gravitational wave forms,...).

 

[madness]

Note that, in my case it isn’t “some posters” arguing, it comes from the literature. There is a lot of variety on this topic, which is an indication that it is mainly just a personal preference. So just find one that you like.

Your reference doesn't show how to find an intertial frame from Newton's law. It gives two approaches, first it says:

Newton's first law deals with non-interacting bodies. It says that the velocity of an isolated body, one removed from the influence of other bodies, is constant. This law defines a set of preferred coordinate frames,
inertial frames, as frames in which Newton's first law holds

This definition requires that we identify "non-interacting bodies". It then goes on to state that the best approximation to an inertial frame is the "fixed stars". Thus, your reference proves my point, i.e. it tells us that we need either 1) Prior knowledge of which frames are intertial (here, the "fixed stars") 2) Knowledge of some bodies that are "non-interacting", or 3) prior knowledge of the kinds of interactions that take place.

I believe you are either mirepresenting or misunderstanding your reference if you claim otherwise.

 

[A.T.]

Your reference doesn't show how to find an intertial frame from Newton's law...
This definition requires that we identify "non-interacting bodies".

And once you identified them, you can find the inertial frame from Newton's law.

 

[madness]

And once you identified them, you can find the inertial frame from Newton's law.

Sure, that's what I wrote already in my post #46 that Dale had quoted. However they can't be identified using Newton's laws alone.

 

[A.T.]

However they can't be identified using Newton's laws alone.

What is the meaning/significance of that "alone" part? Can you give an example of a useful physical law, that doesn't reference some other definitions/concepts. The whole point of general laws in physics is to relate/connect different aspects.

 

[madness]

What is the meaning/significance of that "alone" part? Can you give an example of a useful physical law, that doesn't reference some other definitions/concepts. The whole point of general laws in physics is to relate/connect different aspects.

I have no problem with that. In post #46 I simply noted that there is a minimal set of propositions required to identify intertial frames and make testable predictions. I find it conceptually useful to identify the minimal set(s) required to uniquely specify and empirically measure the relevant physical components of the theory.

 

[Dale]

Sure, that's what I wrote already in my post #46 that Dale had quoted. However they can't be identified using Newton's laws alone.

I don’t see why this is an issue. Every definition requires that you use other definitions too. For this, it is assumed that you know how to identify non-interacting bodies and straight lines and constant velocities. Why are you getting your knickers in a twist about the non interacting bodies and not the other assumptions too? All of those things are part of the law by assumption, without being explicitly defined in it.

A definition is certainly allow to use and reference other concepts that are not defined in the definition. If you couldn’t use other concepts then every definition would be the size of the whole dictionary.

For straight lines you can order a protractor and a ruler from a supply catalog. Examples of straight lines can thus be provided. Similarly for non-interacting bodies, the distant fixed stars are considered an available example of such objects.

 

[madness]

I don’t see why this is an issue. Every definition requires that you use other definitions too. For this, it is assumed that you know how to identify non-interacting bodies and straight lines and constant velocities. Why are you getting your knickers in a twist about the non interacting bodies and not the other assumptions too? All of those things are part of the law by assumption, without being explicitly defined in it.

A definition is certainly allow to use and reference other concepts that are not defined in the definition. If you couldn’t use other concepts then every definition would be the size of the whole dictionary.

For straight lines you can order a protractor and a ruler from a supply catalog. Examples of straight lines can thus be provided. Similarly for non-interacting bodies, the distant fixed stars are considered an available example of such objects.

I simply feel that it is conceptually helpful to lay out all of the necessary statements needed to operationally define the physical terms in the theory. In laying out Newton's laws and then implicitly using additional assumptions in an ad hoc manner we are obscuring the logical foundations of the theory we are using. Maybe others are happy to proceed in this way but it is not my preference.

 

[vanhees71]

Sure, that's what I wrote already in my post #46 that Dale had quoted. However they can't be identified using Newton's laws alone.

Yes you can, that's the achievement of Ludwig Lange mentioned some postings above.

 

[madness]

Yes you can, that's the achievement of Ludwig Lange mentioned some postings above.

Lange's definition uses the term "free particle". His innovation is to use three of them to construct coordinates in three-dimensional space. We still need to know which particles are "free" before we can determine which coordinate systems are inertial.

 

[DrStupid]

We still need to know which particles are "free" before we can determine which coordinate systems are inertial.

Do we really need to know it or is it sufficient to assume it as a working hypotheses? That would imply that that the classification of a frame of reference as inertial or not is provisional as well but that shouldn't be a problem as long as everything remains consistent.

 

[madness]

Do we really need to know it or is it sufficient to assume it as a working hypotheses? That would imply that that the classification of a frame of reference as inertial or not is provisional as well but that shouldn't be a problem as long as everything remains consistent.

If we take any N particles with arbitrary trajectories in some coordinate system, there should* be some transformation of coordinates such that in the new coordinate system each particle follows a straight line with constant velocity. Then we could call those "free" and declare that we have found an "inertial" coordinate system.

*I'm actually unsure of the conditions under which this is true! However I think it gets at the question of how we can operationally determine free particles or inertial frames given only trajectories of particles in one set of coordinates.

 

[DrStupid]

If we take any N particles with arbitrary trajectories in some coordinate system, there should be some transformation of coordinates such that in the new coordinate system each particle follows a straight line with constant velocity. Then we could call those "free" and declare that we have found an "inertial" coordinate system.

If that always works - independent from the choice of particles and their trajectories - what's the problem? Otherwise you need to rethink the assumption that the particles are free.

 

[madness]

If that always works - independent from the choice of particles and their trajectories - what's the problem? Otherwise you need to rethink the assumption that the particles are free.

The problem is that we didn't know which particles were free or not. What if we have M free and N forced particles, but we don't know which are free or forced, rather we only have their trajectories. We can perform coordinate transformations which put the forced ones in straight lines and claim they are free.

 

[DrStupid]

We can perform coordinate transformations which put the forced ones in straight lines and claim they are free.

Again: If that always works, what's the problem?

 

[madness]

Again: If that always works, what's the problem?

The problem is that we may be able to trade off choices of force and inertial frames so that any frame can be called inertial if we choose the forces on the particles appropriately.

 

[DrStupid]

The problem is that we may be able to trade off choices of force and inertial frames so that any frame can be called inertial if we choose the forces on the particles appropriately.

I do not see the problem. We actually do that. In classical mechanics the trajectory of the falling apple is not a straight line and there is a net force acting on it while the trajectory of Newton, sitting unter the tree, is a straight line and there is no net force acting on him. In general relativity it is the other way around.

 

[vanhees71]

Of course you have to assume that an inertial frame exists. Since you can operationally define an inertial frame only if you assume that "free particles" exist, this assumption is implied.

 

[Dale]

I simply feel that it is conceptually helpful to lay out all of the necessary statements needed to operationally define the physical terms in the theory. In laying out Newton's laws and then implicitly using additional assumptions in an ad hoc manner we are obscuring the logical foundations of the theory we are using. Maybe others are happy to proceed in this way but it is not my preference.

OK, so (sticking with the first law only for now) why are you focusing only on the "non-interacting bodies" part and not on the "constant velocity" part? Constant velocity is also not defined in the law but rather is assumed that you can determine if something is moving at constant velocity. There is conceptually no difference between the two. Why does "non-interacting bodies" set off your "ad hoc" and "obscuring logical foundations" alarms but "constant velocity" does not. From a logical standpoint they are not any different.

All systems of definitions (not merely in science, but in math and elsewhere too) inherently either are circular or refer to some undefined term. Science is a little better than most disciplines because a lot can be referred to physical measurement apparatus. E.g. "time is what is measured by a clock" so what is a clock? I don't define it, but instead I can give you instructions for building a clock, or I can give you a reference clock, or I can give you a catalog where you can order a clock. This is the same sort of thing.

 

[madness]

OK, so (sticking with the first law only for now) why are you focusing only on the "non-interacting bodies" part and not on the "constant velocity" part? Constant velocity is also not defined in the law but rather is assumed that you can determine if something is moving at constant velocity. There is conceptually no difference between the two. Why does "non-interacting bodies" set off your "ad hoc" and "obscuring logical foundations" alarms but "constant velocity" does not. From a logical standpoint they are not any different.

I take trajectories to be measurable, i.e. position and its derivatives. Whether a particle is "non-interacting" cannot be measured directly, but only inferred from measurements of trajectories together with the definitions of the theory.