Empirical and Definitional Content of Newton's Laws

[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.

 

[madness]

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.

Which angle? I was referring to their angle(s) in polar coordinates with Earth at the centre. Certainly this doesn't decrease as the distance from Earth increases. We can essentially only measure these two angles (azimuth and elevation). The acceleration of azimuth and elevation goes to zero as the distance of the object goes to infinity, even if we apply linear acceleration here on Earth. As DrStupid says, today we may be able to make more accurate measurements, but only insofar as the stars aren't sufficiently distance to be considered "fixed".

 

[Dale]

Which angle? I was referring to their angle(s) in polar coordinates with Earth at the centre. Certainly this doesn't decrease as the distance from Earth increases.

The angle subtended by their change in position over time. Yes, measured in polar coordinates with Earth in the center.

The acceleration of azimuth and elevation goes to zero as the distance of the object goes to infinity

Yes, but the hypotenuse goes to infinity. Their product is constant and independent of the distance.

 

[madness]

The angle subtended by their change in position over time. Yes, measured in polar coordinates with Earth in the center.

Yes, but the hypotenuse goes to infinity. Their product is constant and independent of the distance.

Sure, but you can't use observations of the night sky to construct an inertial frame using the coordinates of the distant "fixed" stars. Unless you use something like redshift to estimate their radial acceleration.

 

[Dale]

Sure, but you can't use observations of the night sky to construct an inertial frame using the coordinates of the distant "fixed" stars. Unless you use something like redshift to estimate their radial acceleration.

Yes, of course. You definitely need radial information also! If you don't have radial information then how could you apply Newton's laws? You wouldn't have velocities or accelerations but something quite different and not meaningful in the context of the laws.

 

[madness]

Yes, of course. You definitely need radial information also! If you don't have radial information then how could you apply Newton's laws? You wouldn't have velocities or accelerations but something quite different and not meaningful in the context of the laws.

If we have access to detailed measurements of their three-dimensional motion then it ceases to make sense to refer them as being "distant" or "fixed". My impression was that the term "distant fixed stars" was intended to convey something about their usefulness for constructing an inertial frame in contrast to nearby moving objects.

 

[vanhees71]

Today a much simpler frame of reference is the restframe of the cosmic microwave background, which you can establish using local observations. It's very accurately done already with, e.g., the CMBR satellites WMAP and Planck.

 

[Dale]

If we have access to detailed measurements of their three-dimensional motion then it ceases to make sense to refer them as being "distant" or "fixed". My impression was that the term "distant fixed stars" was intended to convey something about their usefulness for constructing an inertial frame in contrast to nearby moving objects.

The term “distant fixed stars” is just a historical term. I agree that the term doesn’t make sense given modern astronomical knowledge. But for historical reasons it is still used when discussing constructing an inertial reference frame from astronomical objects. I believe that it was originally intended to refer to any actual stars (besides the sun), as opposed to planets and comets which were seen as wandering stars rather than fixed stars.

 

[f todd baker]

My goodness, this is a long chain of discussion! I haven't read it all but what I did read did not mention something which I always taught my intro students in the first class. Newtonian physics requires that we somehow understand intuitively without derivation four concepts: length, time, force, and mass. Force and mass are the tricky ones for students, force is a push or pull and mass somehow measures how much stuff you have. But suppose that we come up with an operational definition of mass--a kilogram is the mass of some standard chunk of stuff in some vault in Paris. And suppose we imagine that, although we do not have an operational definition of force, we can imagine having a machine (maybe a spring) which will reliably exert a constant force; then we can imagine doubling the force (two machines), tripling it, etc. Now we interact with the real world and do an experiment and easily discover that a∝F/m. We make this into an equation by adding a proportionality constant C which we can choose to be anything we want because F has not been defined. I choose C=1 and voila, F=ma and F is now operationally defined as the force which causes a 1 kg mass to have an acceleration of 1 m/s^2.
But, although that is what we like to do as physicists, it is not unique because F and m are not unrelated. The other way to approach the problem is to choose F rather than m to build our system of units. This is exactly what the Imperial units do, based on the pound, foot, second rather than kilogram, meter, second. Experiment still finds a∝F/m and I still choose C=1 and I still have the same Newton's second, m=a/F, and now a unit of mass is the mass which will experience an acceleration 1 ft/s^2 if pushed with 1 lb of force. (I realize that the more conventional Imperial unit for mass is the mass of a 1 lb weight, but then the second law is no longer F=ma. My goal here is to illustrate that one only needs three intuitive concepts to start physics, F and m not being independent.)

 

[vanhees71]

Well, yes, that's of course a good way to start, but we are discussing the problem that for doing all this you need to start with some spacetime model. It all starts with "kinematics" before it comes to "dynamics", and the issue with the Newtonian system of postulates was known from the very beginning, i.e., how to operationally determine either Newton's absolute space and absolute time (Newtonian point of view) or a set of bodies defining an arbitrary inertial reference frame (Leibnizian point of view), but we've discussed this above in great detail already.

Of course FAPP your approach is the right one and the only way to get started in physics.

 

[avicenna]

First, we must know that our textbook explanation of Newton's laws is our accepted approach which may be different in some ways to Newton's original; e.g. space and frame.
1) The Latin of the Principia is "Three Axioms" or accepted truths to start with, not laws as with testable laws.
2) mass - The Principia also assumed mass as a defined concept/axiom using the weighing scale and the pendulum to support its concept of mass.

The first law defines the inertial reference frame. A "perfect" orbiting space lab with no rotation with respect to the Earth is an inertial reference frame. An object can be made to stay still or move with uniform motion. The inertial reference frame is only an ideal and never testable to an absolute precision.

The second law is an axiom/definition of force. I think only within an inertial reference frame.

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 third law need no testing as it is a consequence of the 1st and 2nd law.

 

[Trying2Learn]

I am not sure if I can help, as most of the time, I ask questions.

Still, this confused me too, until I learned that aside from the three laws, Newton also put forth some definitions, BEFORE the laws.

Here are four
The quantity of matter is the measure of the same, arising from its density and bulk, conjointly.

The quantity of motion is the measure of the same, arising from the velocity and the quantity of matter, conjointly.

The force is a power of resisting, by which every body, as much as in it lies, continues in its present state, whether it be be of rest

An impressed force is an action exerted upon a body, in order to change its state, either of rest, or of uniform motion in a right line.

Then, I do remember reading somewhere (I do not know where) that some physicists feel that the first law should not be a law, but a definition (or something). I do not recall. This is probably not much help.

 

[A.T.]

The third law need no testing as it is a consequence of the 1st and 2nd law.

How does the 3rd law follow from the 1st and 2nd?

 

[avicenna]

How does the 3rd law follow from the 1st and 2nd?

If m₁ acts on m₂ with a force f, then f=m₂ a₂; a₂ is relative to m₁. Then m₁ too has relative acceleration a₂ relative to m₂. Thus there is a force acting on m₁; the magnitude of the force is also the same f = m₂ a₂.

I think Newton's law applies only in inertial reference frames; it cannot be otherwise. We assume that m₁ and m₂ are initially at rest in an inertial reference frame. Then a force f appears.

 

[Dale]

The third law need no testing as it is a consequence of the 1st and 2nd law.

This is certainly not correct. In fact, it is the third law that contains the empirical content of Newton’s laws.

If m₁ acts on m₂ with a force f, then f=m₂ a₂; a₂ is relative to m₁.

No, a₂ is relative to any inertial frame. The acceleration with respect to m₁ may be very different from a₂

 

[DrStupid]

In fact, it is the third law that contains the empirical content of Newton’s laws.

And that's why Newton supported it with experimental evidence.