Empirical and Definitional Content of Newton's Laws

[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.

 

[Dale]

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.

Trajectories are only measurable because you accept the validity of the relevant measurement devices and because you refer the output of those devices to some geometrical theory. Without accepting those devices and theories the trajectories are not measurable either. There is really no in-principle difference between that and the distant fixed stars and Newton's laws.

You really should look into the Newton-Cartan approach. The more you write the more I think it would be satisfying to you. In that approach whether a particle is "non-interacting" can be measured directly using an accelerometer. In fact, it is a far more direct measurement than trajectories are.

 

[madness]

Trajectories are only measurable because you accept the validity of the relevant measurement devices and because you refer the output of those devices to some geometrical theory. Without accepting those devices and theories the trajectories are not measurable either. There is really no in-principle difference between that and the distant fixed stars and Newton's laws.

I agree that there are some assumptions required to empirically measure position and velocity. I can also see how Newton's fixed stars can be taken a priori as a reference frame. But I don't see how the ability to identify "non-interacting" particles could be framed as a basic premise, unless we attribute that property to a well-known object like the sun that we can all agree on. How else would the theory allow us to find this object?

You really should look into the Newton-Cartan approach. The more you write the more I think it would be satisfying to you. In that approach whether a particle is "non-interacting" can be measured directly using an accelerometer. In fact, it is a far more direct measurement than trajectories are.

Perhaps I should. I'll try to find some time for that soon.

 

[Dale]

I agree that there are some assumptions required to empirically measure position and velocity. I can also see how Newton's fixed stars can be taken a priori as a reference frame. But I don't see how the ability to identify "non-interacting" particles could be framed as a basic premise, unless we attribute that property to a well-known object like the sun that we can all agree on. How else would the theory allow us to find this object?

If you allow the sun then why not the distant fixed stars?

 

[madness]

If you allow the sun then why not the distant fixed stars?

All of them or just one of them? It would be tricky to claim that they are all non-interacting, as we'd have to nonlinearly transform coordinates into a system where they all travel in straight lines at constant speeds.

 

[vanhees71]

True, but of course also the functioning of the accelerometer needs to use the fundamental laws. At the end it's a question of consistency and the limits of accuracy you choose to achieve or are able to achieve. You cannot get out of this dilemma that you use the very laws to construct measurement devices you want to test using these devices. At the end it's a question of consistency whether a model like Newtonian mechanics is a valid model to describe the observed phenomena or not, and you cannot even start to measure anything without an assumption of such a model.

Indeed, without much thinking usually you start to use the Earth as a refrence body. Say you want to measure the laws of free fall (because it's clear that there's always the gravitational interaction of anybody with the Earth; the unavoidable gravitational interaction between the bodies can usually be neglected, but that's another issue, but it's also an issue of accuracy). So you take the Earth as the reference body and take a ruler to measure vertical distances ("vertical" defined as the direction of the gravitational force/acceleration, which we can assume as homogeneous for not too large spatial regions we perform our measurements in; from Newton's Law of gravity we know the relevant length scale here is of the order or the radius of the Earth; so in usual lecture-hall experiments it's safe to assume that the gravitational acceleration $\vec{g}$ is just a constant). Now you realize that anybody falls in a straight line along this ruler. You can measure distances with this ruler (say from the starting point of the falling body to the floor of your lab), because you assume (!) the validity of Euclidean geometry for the physical space.

Now you also need a clock. One way is of course to use a mathematical pendulum (or to be more accurate Huygens's isochronous pendulum with a body moving along a cylcloid to avoid a dependence on the amplitude). Newton's Laws (which are again assumed here!) lead to a measure of time due to the formula $\omega=\sqrt{g/l}$, for which you only need to be able to measure the length $l$ of the pendulum. The value of $g$ is not so important, because you can take it as a constant around you lab, and you can simply define a measure of absolute time by taking the frequency of a certain standard length to define your unit of time. Of course you can now check the law for different pendulums by measuring the frequency as a function of $l$ using such a standard clock to compare the frequencies of these various pendulums.

Now we have established a measure of time. Now of course you can measure the trajectory of bodies as a function of the so quantified time $t$. Of course, you'll then find with more or less accuracy, the assumption $\vec{g}=\text{const}$ confirmed when measuring freely falling bodies has a function of the height $x=h-g t^2/2$ (if you always start with 0 initial velocity).

As a next step you can also check the law for a body moving not only along the "vertical". If you neglect air resistance of course you get the law that velocity components of moving bodies perpendicular to the direction of $\vec{g}$ stay constant. This pretty much ensures you that taking the Earth as a reference body is a good approximation of an inertial frame, where of course you have to take into account the gravitational interaction between the observed bodies and the Earth.

There are of course a lot of assumptions and some math going into it like solving the equations for free(ly falling) bodies or a mathematical pendulum. Then you can make all kinds of measurements and check whether all this assumptions stay consistent with each other and in accordance with the predictions using Newtonian mechanics.

As stressed before, it's also a question at which accuracy you measure. Of course we all know that the Earth is not really a reference point to define an inertial frame. The Foucault pendulum experiment demonstrates that it is a rotating frame as the correct prediction of the precession of the pendulum's plane of oscillations assuming the corresponding corrections due to the inertial forces (here the Coriolis force is sufficient) expected in a uniformly rotating reference frame (i.e., the Earth rotation around its axis).

Also the assumption that the bodies are not mutually interacting is of course only an approximation, and indeed Cavendish managed to demonstrate Newton's universal gravitational law and to measure with some accuracy Newton's universal gratvitational constant with his experiment torsion balance using the mutual gravitational interactions of bodies (at an accuracy within about 1% compared to the modern value).

Of course, as an empirical science physics cannot be axiomatized. It's always an interplay between theoretical thoughts and the construction of measurement devices to make quantitative observations. The "natural laws" are always subject to tests and consistency checks with the underlying theoretical assumptions.

The history of physics shows that indeed adjustments to the very fundamental laws are happening, though in an amazingly slow rate. Newton's mechanics was consistent for around 200-300 years until the spacetime model finally turned out to be only an approximation, when in 1905 Einstein discovered special relativity and thus introduced the more comprehensive Minkowski-space spacetime model, which was consistent with a larger realm of phenomena, i.e., not it was consistent with both mechanics and electrodynamics. Just 10 years later the spacetime model had to be adjusted again, because it turned out that gravity could be most easily described by reinterpreting it as a curved spacetime manifold and at once unified the phenomenon of inertia and gravitational forces (though only in a local sense of course). Though General Relativity (GR) up to now stands to be highly accurate, it may well be that one day one needs even more accurate laws, and that's why GR is also tested with ever more accurate measurements as the advance of technology allows.

 

[madness]

You cannot get out of this dilemma that you use the very laws to construct measurement devices you want to test using these devices.

One would hope that we could specify a set of measurement "primitives" along with a set of definitions that link those measurements to abstract (i.e., non-measurable) physical quantities, and then analyse that system to ascertain whether these physical quantities can be uniquely determined from measurements of the "primitives". In my case the primitives were chosen to be position and time, whereas the abstract quantities were force, intertial frame, mass etc. Ultimately, testing the theory can only be done on the basis of directly measurable quantities, in which case this debate about whether we can truly identify inertial frames and non-interacting bodies may become irrelevant from an observational perspective.

 

[DrStupid]

It would be tricky to claim that they are all non-interacting, as we'd have to nonlinearly transform coordinates into a system where they all travel in straight lines at constant speeds.

Or you just assume them to be at rest relative to each other. That's what Newton was sure of and it was sufficient within the accuracy of measurements that time. Today we know it is not the case but we could use the cosmic background radiation instead.

 

[madness]

Or you just assume them to be at rest relative to each other. That's what Newton was sure of and it was sufficient within the accuracy of measurements that time. Today we know it is not the case but we could use the cosmic background radiation instead.

I'm not sure I get it. They appear to stay still in our frame of reference regardless of whether we are moving or accelerating. I can't see how to get an inertial frame from that.

 

[Dale]

All of them or just one of them? It would be tricky to claim that they are all non-interacting, as we'd have to nonlinearly transform coordinates into a system where they all travel in straight lines at constant speeds.

As long as you are working with time scales shorter than a few centuries they do travel in straight lines at constant speeds. And if you use the Newton-Cartan approach then they travel on geodesics at any time scale.

They appear to stay still in our frame of reference regardless of whether we are moving or accelerating.

No, that definitely isn't true. If you are accelerating then the distant fixed stars appear to fall.

 

[madness]

No, that definitely isn't true. If you are accelerating then the distant fixed stars appear to fall.

If they were fixed they wouldn't fall. The relative speed and acceleration of a point at infinity would be zero for any linear motion. Perhaps you mean rotation?

Edit: I suppose I'm referring to changes in angle on the sky and size of the object, which is how I imagined it's motion would be inferred.

 

[jbriggs444]

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.

We have rich history outside the context of Newton's laws where measures are taken to assure that particles are free from external influences.

 

[Dale]

If they were fixed they wouldn't fall.

Yes, if they are fixed then you are using an inertial frame and they won’t fall.

 

[madness]

We have rich history outside the context of Newton's laws where measures are taken to assure that particles are free from external influences.
View attachment 267641

Right, and this is done using prior knowledge or assumptions about the kinds of influences that occur on particles.

Yes, if they are fixed then you are using an inertial frame and they won’t fall.

They are approximately fixed because they are sufficiently distant, and this is true regardless of the linear acceleration we apply here on Earth (i.e., it's true even in what we would consider non-inertial frames).

 

[Dale]

this is true regardless of the linear acceleration we apply here on earth

No. If you use an accelerating frame they will not be fixed, they will be accelerating. That is what I already said was wrong above.

 

[Ibix]

No. If you use an accelerating frame they will not be fixed, they will be accelerating. That is what I already said was wrong above.

Trivial example: an Earth centered rotating frame. Distant stars move in circles around us, not remaining fixed above an observer at rest in such a frame.

 

[madness]

Trivial example: an Earth centered rotating frame. Distant stars move in circles around us, not remaining fixed above an observer at rest in such a frame.

I said linear acceleration, not rotation. We can only measure the size and angle on the sky of each star, in which case stars at a sufficient distance are fixed regardless of any linear acceleration we apply here on earth.

 

[atyy]

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?

The second law is the definition of a force, and is empty without further specifying a functional form for the force, such as the law of universal gravitation.
The first law can be seen as a special case of the second.
The third law specifies that forces have a certain symmetry in inertial frames. The third law does not hold for forces that can be defined in noninertial frames.
The choice of definitions is a good one by experience. Notably the third law fails in relativity, and has to be generalized to momentum conservation. Noether's theorem relates conservation laws and symmetries.

 

[Dale]

I said linear acceleration, not rotation. We can only measure the size and angle on the sky of each star, in which case stars at a sufficient distance are fixed regardless of any linear acceleration we apply here on earth.

As the distance increases the measured angle decreases but the hypotenuse increases. If you work it out you will find that the measured linear acceleration of distant objects is independent of the distance. If you are using a non-inertial frame then the distant stars indeed fall, they do not stay fixed.

 

[DrStupid]

The second law is the definition of a force, and is empty without further specifying a functional form for the force, such as the law of universal gravitation.

No, it is not empty without force laws. If you have a frame of reference that you accept to be inertial (e.g. a locally free falling frame of reference) and there is a body with the mass m and the acceleration a then you do not need any force law to know that the force F=m·a is acting on this body. Together with the third law you also know that there must be a force acting on at least one other body and that the sum of all other forces is -m·a. That is way more than nothing.

It is the other way around: Force laws are empty without the laws of motion. The universal law of gravitation or any other force law don't tell you anything without the laws of motion and it would be impossible to derive force laws from expertimental observations without the laws of motion. The laws of motion are the basis that all force laws are standing on.

The third law does not hold for forces that can be defined in noninertial frames.

I would say that something that does not comply with the laws of motion (e.g. fictitious forces) is not a force. In that sense there can't be any forces that the third law doesn't hold for. But maybe this is a matter of taste. It seems even Newton himself was not happy with the restriction of the laws of motion to interactive forces.

 

[DrStupid]

As the distance increases the measured angle decreases but the hypotenuse increases.

And how do you measure the hypotenuse? Today it is is quite easy to measure even small accelerations of distant stars. But more than three centuries ago it was impossible. Thus the fixed stars were just a theoretical reference for an inertial frame without any practical relevance. I think this is what madness means.