What Do Newton's Laws Say When Carefully Analysed

[kith]

Newtonian Gravity?

It depends on what "mediated by a field" is supposed to mean. Sure, you can formally introduce a field quantity $\vec G$ and write Newton's law of gravity as $\vec F = m \vec G$. This has the same structure as the corresponding law in electrodynamics but as @vanhees71 notes, Newtonian Gravity doesn't involve the field as a dynamical physical system which is what's usually meant by "mediated by a field".

 

[DrStupid]

Sorry @DrStupid, I will stick with Dr Calkin here. His approach seems much clearer than yours.

It seems so because you accept the limitation of Newton I to non-interactin bodies as a matter of course. You think it is just there - coming from nowhere - because it is commonly used. But that's not as self-evident as you think. It would have been possible to define forces without restriction to interactions. I already mentioned that fictitious forces are commonly used as well. You should ask yourself where the consensus, not to assume them to be forces, comes from. The answer is Newton III.

Per the cited approach, since you have a non-interacting particle which is accelerating then your reference frame is non inertial. That seems the most direct approach to me.

Is seems but it isn't. Newton I deals with particles in presence or absence of forces - not with presence or absence of interactions. That's not necessarely the same. You need to conclude from one to the other. The approach is no longer as direct as it looks like if you don't skip this step.

Sure, you can go out of your way to use a poor formulation of Newton’s laws which requires you to use all of the laws for even the simplest cases, but you certainly don’t have to and I personally don’t choose to.

On what basis are you rating Newton's formulation of the laws of motion as "poor"? What makes a formulation that needs additonal assumptions - not included in the laws of motion - better?

Even if your favorit formulation appears to be better (for what criterion ever) - why are you using it in this thread? The topic is "What Do Newton's Laws Say When Carefully Analysed". Starting with laws of motion that are not equivalent with Newton's laws of motion (resulting in different conclusions) is far from beeing careful. You shouldn't even use Newton's name for it.

And talking about poor formulations: Claiming "Newton's first law deals with non-interacting bodies." (just to take an example) is at least mistakable. Newton I says:

"Every body perseveres in its state of being at rest or of moving uniformly straight forward, except insofar as it is compelled to change its state by forces impressed."

Interacting bodies are obviously included in Newton I - even if you take the relation between interaction and forces as given. And no, this is not an outdated or unpopular version of the first law. You will find it with different wordings but same content everywhere in the scientific literature. If you claim Newton's original formulations of the laws of motion to be "poor" you should at least apply the same standards to your sources as well.

 

[DrStupid]

You have to establish an inertial reference frame first, according to the definition of Newton I.

If that means "according to the definition of Newton I only", than no I don't. I already explained why this is not even possible.

Then you assume that particles far away from any other particles are "force-free" and check whether all such particles move in rectilinear uniform motion relative to each other.

How do I assume that without using Newton III?

 

[olgerm]

An inertial frame is a reference frame where all non interacting particles travel in straight lines at constant velocity.

This only works if do not count inertial forces as forces. But if you define force by $F=a*m$, then inertial forces would be forces.
But using Newtons 3. law would not solve problem, because in uniformly accelerating frame all particles have the same inertial force.

 

[Dale]

This only works if do not count inertial forces as forces

That is why the approach of Dr Calkin is nice. Inertial forces do not come from interactions. So by focusing the definition of an inertial frame on non-interacting bodies then you automatically and naturally exclude inertial forces without even having defined forces at that point.

 

[atyy]

This only works if do not count inertial forces as forces. But if you define force by $F=a*m$, then inertial forces would be forces.
But using Newtons 3. law would not solve problem, because in uniformly accelerating frame all particles have the same inertial force.

Newton's 3rd law helps because inertial forces are not part of action-reaction pairs.

 

[Jimster41]

“Naturally” seems a stretch. Definitions are nice but... I’m having a hard time imagining a real non-interacting body I’m somehow checking for rectilinear motion...

 

[atyy]

“Naturally” seems a stretch. Definitions are nice but... I’m having a hard time imagining a real non-interacting body I’m somehow checking for rectilinear motion...

For interacting bodies (in the presence of Newtonian gravity, there is no such things as a non-interacting body), one must use all 3 laws (and maybe a bit more, I'm not sure).

If you look at posts #22 by me, #29 by @Demystifier, #45 by @DrStupid they essentially make the similar point that additional content beyond F=ma is needed. The concept of a pre-defined non-interacting particle is one way to provide the additional content. However, if there are no non-interacting particles, then one must specify the form of the forces. In fact, this is the point of the OP - if N2 is taken as a definition, and N1 is a special case of N1, then N2 and N1 alone are physically empty.

Also, in post #55 by @bhobba he points out that nowadays we define inertial frames through the symmetry of the laws, which is consistent with @DrStupid's preference to include N3, since N3 is conservation of momentum in the Newtonian framework, which is equivalent to a symmetry via Noether's theorem.

 

[Dale]

I’m having a hard time imagining a real non-interacting body I’m somehow checking for rectilinear motion.

I agree. To me, this is the weakness of Calkin’s approach. Not that it somehow implicitly uses or hides N3 in the definition of an inertial frame, but rather that it is not practical.

 

[Dale]

Newton I deals with particles in presence or absence of forces - not with presence or absence of interactions

Calkin disagrees, as do I.

The reason that you are going around in circles with me here is that you keep trying to argue based on premises that I don’t accept. I like Calkin’s formulation of Newton’s laws precisely because of how clean it is on this topic (inertial frames defined by first law alone). You keep explaining why the original formulation is not clean. That only re-convinces me to not use the original formulation.

Do you have any argument showing that Calkin’s formulation of Newton’s laws (not other formulations) requires N3 to define an inertial frame? It doesn’t seem necessary to me with his approach.

Starting with laws of motion that are not equivalent with Newton's laws of motion (resulting in different conclusions) is far from beeing careful. You shouldn't even use Newton's name for it.

It isn’t me using Newton’s name, it is Calkin. He felt that his formulation was equivalent enough to be given Newton’s name, and I agree. Complain as you will, but in science it is expected that later authors may reformulate seminal works. It is common. The seminal authors get the first word, but not the last word. Newton isn’t the pope and his words are not canonized.

Calkin has his right to reformulate Newton’s laws as he saw fit. The formulation is empirically equivalent, so the “different conclusions” are ok. Indeed, everyone mentioning the derivation from symmetry is making a similar deviation from Newton’s formulation since he explicitly included an undetectable absolute space and time as part of his original formulation. Are you objecting to their reformulation? No, nor should you; it is OK for them to do it as well as it is for Calkin.

 

[vanhees71]

It depends on what "mediated by a field" is supposed to mean. Sure, you can formally introduce a field quantity $\vec G$ and write Newton's law of gravity as $\vec F = m \vec G$. This has the same structure as the corresponding law in electrodynamics but as @vanhees71 notes, Newtonian Gravity doesn't involve the field as a dynamical physical system which is what's usually meant by "mediated by a field".

It has the same structure as the Coulomb Law of electrostatics, not electrodynamics! That's the distinction between an action-at-a-distance-approximation, which is well justified for non-relativistic motions of not too far distant charges, where radiation reaction can be neglected, and a real field theory with the fields as dynamical entities in addition to the mechanical objects (here "point particles").

 

[vanhees71]

If that means "according to the definition of Newton I only", than no I don't. I already explained why this is not even possible.
How do I assume that without using Newton III?

What has Newton III to do with that? Newton III says nothing about the range of interactions.

It's of course true that this apparently simply foundations are more subtle than one thinks. E.g., the gravitational interaction is not easy to come by, because it's long-ranged (i.e., the interaction potential goes only with $1/\text{distance}$ and it cannot be "screened", and then the asymptotic free motion is not uniform motion at all. Here, of course the equivalence principle (which applies to Newtonian gravity too) comes to a rescue: As far as tidal forces can be neglected (and their potential goes faster to 0 than with $1/\text{distance}$!) a free-falling rest frame is at least locally an inertial frame, and for this argument you indeed need Newton III (or momentum conservation and the associated center-mass law, which follows from homogeneity of space and Gailei-boost invariance, i.e., uses Newton III).

As I said, you need all three postulates (+ the tacit assumption that space for any inertial observer, and in Newton's absolute space and absolute time model that implies all observers, even accelerated ones) to make the theory complete. Nevertheless, the logic is already right in Newton's original approach: First you need to establish what inertial frames are (Newton I), then what forces are (Newton II), and finally Newton III. Of course a more systematic mathematical way is the modern one based on symmetries, and in this case it's the symmetries of the space-time model, which is a fibre bundle (identical copies of Euclidean affine manifolds along the "directed time axis").

 

[vis_insita]

I believe many of the issues discussed here can be illuminated by a covariant formulation of Newtonian Mechanics, in which all laws of motion are given independently of any reference frame. The basic idea is the following: Contrary to “velocity” which can be defined on any manifold, in order to define "acceleration" you need a method to compare velocities at different events. This requires a covariant derivative. Such a covariant derivative also defines “straightness” in space and time, which is the natural interpretation of the “uniform motion” the First Law is talking about. Thus, one needs to define acceleration as the deviation from uniform motion as $$\nabla_{\boldsymbol{u}}\boldsymbol{u}=\boldsymbol{a},$$ just like in General Relativity. The only difference is in the conditions one has to impose on $\nabla$ in order for it to be compatible with Newtonian space time.

One such requirement is that “Absolute, true and mathematical time, of itself, and from its own nature flows equably without regard to anything external.” This “equable flow” has a nice formal interpretation as $\nabla\mathrm{d}t=0$. It implies that if a vector field $\boldsymbol{x}$ is spatial i.e. $\langle\mathrm{d}t,\boldsymbol{x}\rangle=0$, then its covariant derivative is too (which will be important below), i.e. $$\langle\mathrm{d}t,\nabla_{u}\boldsymbol{x}\rangle=\nabla_{u}\langle\mathrm{d}t,\boldsymbol{x}\rangle-\langle\nabla_{u}\mathrm{d}t,\boldsymbol{x}\rangle=0.$$

Another requirement is that absolute space, defined by $t=\text{const.}$, has to be flat with a euclidean structure. (This means the curvature calculated from $\nabla$ fulfills some identities, but they are not important in the following.) With this introduction my preferred interpretation of the First Law is this:

“1st Law: There exists a covariant derivative $\nabla$ that is compatible with the space time structure in the above sense. Isolated, ie. non-interacting, particles follow geodesics in space time $\nabla_{\boldsymbol{u}}\boldsymbol{u}=0.$"

I have not defined “non-interacting”, yet. But this definition will be made independent of the Third Law below. In any case, this interpretation of the First Law enables the following definition of inertial frames, which is completely independent of forces, and only relies on $\nabla$.

An inertial frame is defined by a set of three spatial orthogonal vectors $\boldsymbol{e}_{i}$ which are parallel transported along a straight line through space time. The tangent $\boldsymbol{e}_{0}$ of that line has to fulfill the condition $\langle\mathrm{d}t,\boldsymbol{e}_{0}\rangle=1$, so that the curve described by the origin is parameterized by absolute time. This means an inertial frame is defined as a set of vectors $\boldsymbol{e}_{0},\boldsymbol{e}_{i=1,2,3}$, with

$$\langle\mathrm{d}t,\boldsymbol{e}_{0}\rangle =1$$
$$\langle\mathrm{d}t,\boldsymbol{e}_{i}\rangle =0$$
$$\boldsymbol{e}_{i}\cdot\boldsymbol{e}_{k} =\delta_{ik}$$

and the two dynamical conditions

$$\nabla_{\boldsymbol{e}_{0}}\boldsymbol{e_{0}}=0,\qquad\nabla_{\boldsymbol{e}_{0}}\boldsymbol{e}_{i}=0$$

(Because of the “equable flow of time” the spatial basis vectors remain spatial throughout, for any system.) The first of the dynamical condition is the precise meaning of the coordinate system being non-accelerating in its linear motion. The second means it is non-rotating.

Now it appears clear (I believe) that the definition of inertial frames did not rely on the notion of "interaction". We only used “straightness” in space and time. But of course in reality we cannot know if a particle actually moves along a straight line through space time, independently of knowing whether it is interacting or not. But this is not a logical or philosophical problem. It only means that in the natural sciences all theoretical postulates have to be interpreted in terms of each other for practical applications. This is due to the double nature of scientific law, as simultaneously defining its basic notions and making nontrivial statements about them. This is no peculiarity of Newtonian Mechanics.

To solve this issue practically one just has to hypothesize what force laws describe the possible interactions on the particle, to assess whether it can be judged as “non-interacting” in any particular situation. You do not need a criterion that applies to all possible situations. On the contrary, one is completely free in postualting such laws. The Second Law can thus be interpreted as

“2nd Law: For every particle (of mass $m$) there exists a spatial vector field $\boldsymbol{F}$ on space time, such that the particle's equation of motion is $m\nabla_{\boldsymbol{u}}\boldsymbol{u}=\boldsymbol{F}$.”

In this formulation the First Law is not a special case of the Second Law. The First Law postulates the existence of a covariant derivative, and the Second Law postulates the existence of a vector field. Only the second part of the First Law, concerning the motion of isolated particles, can be made a special case of the Second Law, but this is also the less important part.

The formulation of both Laws is completely covariant. Since they are indepenent of frames of reference, they are not restricted to inertial frames. In particular, this means that the vector field $\boldsymbol{F}$ only describes interactions. You may not add any “fictitious forces” on a whim. Fictitious forces only ever enter the picture when expressing the equation of motion with respect to a reference frame which is non-inertial in the above defined sense.

You also don't have to use any auxilliary criterion to distinguish real forces from fictitous forces. Any vector field is a possible real force, even those that don't satisfy the Third Law. Fictitious forces cannot be regarded as a vector fields in space time, since they are related to the coefficients of $\nabla$ in an arbitrary reference frame and thus do not transform like vectors.

To see how they come about let's just introduce an orthogonal frame S of spatial vectors $\boldsymbol{e}_{i}$ along a curve in space time, which fulfills neither of the dynamical constraints on an inertial frame, i.e. we have

$$\nabla_{\boldsymbol{e}_{0}}\boldsymbol{e}_{0}=\Gamma_{00}^{i}\boldsymbol{e}_{i}$$

(There still is no $\boldsymbol{e}_{0}$-component, because $\langle\mathrm{d}t,\boldsymbol{e}_{0}\rangle\equiv 1$, implying $\Gamma_{00}^{0}=0$.) And

$$\nabla_{\boldsymbol{e}_{0}}\boldsymbol{e}_{i}=\Gamma_{i0}^{k}\boldsymbol{e}_{k}.$$

(All other $\Gamma$ may be assumed to vanish.) The tangent $\boldsymbol{u}$ to the particle worldline and $\boldsymbol{e}_{0}$ are related by the follwing covariant version of the “Galileian velocity-addition formula”

$$\boldsymbol{u}=\boldsymbol{v}+\boldsymbol{e}_{0},$$

where $\boldsymbol{v}$ is the velocity of the particle relative to the origin, which may be equivalently defined as

$$\boldsymbol{v}=\nabla_{\boldsymbol{e}_{0}}\boldsymbol{r},$$

the derivative of the spatial position vector $\boldsymbol{r}$ from the origin to the particle. Now, if the frame rotates with anguar velocity $\boldsymbol{\omega}$ we have

$$\nabla_{\boldsymbol{e}_{0}}\boldsymbol{e}_{i}=\boldsymbol{\omega}\times\boldsymbol{e}_{i}$$

Also defining

$$\nabla_{\boldsymbol{e}_{0}}\boldsymbol{e}_{0}=\boldsymbol{b}$$

which is a spatial vector, we obtain the following form of the Second Law

$$\boldsymbol{F}=m\nabla_{\boldsymbol{u}}\boldsymbol{u}=m\nabla_{\boldsymbol{e}_{0}}(\boldsymbol{v}+\boldsymbol{e}_{0})=m\nabla_{\boldsymbol{e}_{0}}\boldsymbol{v}+m\boldsymbol{b}.$$

The first term on the right hand side may be evaluated further using

$$\boldsymbol{v} = \nabla_{\boldsymbol{e}_{0}}\boldsymbol{r}=\dot{r}^{i}\boldsymbol{e}_{i}+\boldsymbol{\omega}\times\boldsymbol{r}$$

which gives

$$\nabla_{\boldsymbol{u}}\boldsymbol{u}=\ddot{r}^{i}\boldsymbol{e}_{i}+\boldsymbol{\omega}\times(\dot{r}^{i}\boldsymbol{e}_{i})+\boldsymbol{\omega}\times(\dot{r}^{i}\boldsymbol{e}_{i}+\boldsymbol{\omega}\times\boldsymbol{r})+\nabla_{e_{0}}\boldsymbol{\omega}\times\boldsymbol{r}+\boldsymbol{b}.$$

Now, I define the following “derivatives w.r.t fixed axes of S”

$$\boldsymbol{\ddot{r}_{S}}=\ddot{r}^{i}\boldsymbol{e}_{i}$$
$$\boldsymbol{\dot{r}_{S}}=\dot{r}^{i}\boldsymbol{e}_{i}$$

The dots are only meant as suggestive notation. These are not the derivatives of any vectors, in particular $\boldsymbol{\dot{r}_{S}}$ is not equal to the relative velocity defined above $\boldsymbol{\boldsymbol{\dot{r}_{S}}}\neq\boldsymbol{v}=\boldsymbol{\boldsymbol{\dot{r}_{S}}}+\boldsymbol{\omega}\times\boldsymbol{r}$. The Second Law thus reads

$$m\boldsymbol{\ddot{r}_{S}}=\boldsymbol{F}-2m\boldsymbol{\omega}\times\boldsymbol{\dot{r}_{S}}-m\boldsymbol{\omega}\times\left(\boldsymbol{\omega}\times\boldsymbol{r}\right)-m\boldsymbol{\dot{\omega}}\times\boldsymbol{r}-m\boldsymbol{b},$$

where now the right hand side contains all the usual fictitious forces. But those are no more “fictitious” than $m\boldsymbol{\ddot{r}_{S}}$ on the left hand side. Both are just components of the absolute (covariant) acceleration of the particle. If one reviews the usual elementary treatments of “accelerated frames”, I think this is exactly what they do to derive the formulas. Only they are not talking about “covariant derivatives”, but more vaguely of “derivatives with respect to axes that are fixed in space”. But this is the same thing if “fixed in space” means $\boldsymbol{\omega}=0$ and $\boldsymbol{b}=0$.

At no point did I have to use the Third Law and still there was no logical problem defining inertial frames or formulating the First Law.

By the way it is natural to include Newtonian Gravity as curvature of space time, by stating the equivalence principle in the form "inertial motion = free fall". Then the Second Law formally holds as stated above. Only $\boldsymbol{F}$ is also free of gravitational forces, which are instead included in $\nabla$ and are in this sense more on par with inertial forces. Also all "inertial frames" become "local inertial frames".

 

[Dale]

a covariant formulation of Newtonian Mechanics, in which all laws of motion are given independently of any reference frame

Welcome to PF @vis_insita , what an excellent first post!

Do you have a reference for this approach? Also, since F is a vector field on spacetime then how do you introduce the third law?

 

[DrStupid]

Do you have any argument showing that Calkin’s formulation of Newton’s laws (not other formulations) requires N3 to define an inertial frame?

I need to know that formulation to answer this question. In the source you linked above I can see a wording for Lex III but not for I or II. There are a lot of statements about them, but I do not find the laws themselves. Maybe I just missed them. Could you please post the exact wordings?

 

[DrStupid]

Newton III says nothing about the range of interactions.

That's not the point. The range of interactions doesn't matter if forces would need no interactions at all. How do you exclude that without Newton III?

 

[Jimster41]

That was clear. #103. In fact I rather enjoyed that math.

It reinforces what @vanhees71 said about Newton's assumption re Euclidean background space-time.

It also clarifies what bothers me... in what space-time does this co-variant derivative exists? It feels like an "arbiter of all Newtonian dilemmas" but I don't get what gives it reference, how does it avoid being accused of being a "preferred frame" or in some sense occupying one, like some background geometry?

Unless the idea of the third law is to say that the covariant derivative is a benign and strictly helpful artifice because it always treats any two frames it connects as equals (forces are always equal and opposite). IOW, it's reference is only physically defined via interactions - for which no frame is considered preferred?

 

[Jimster41]

The formulation of both Laws is completely covariant. Since they are indepenent of frames of reference, they are not restricted to inertial frames. In particular, this means that the vector field FF\boldsymbol{F} only describes interactions. You may not add any “fictitious forces” on a whim. Fictitious forces only ever enter the picture when expressing the equation of motion with respect to a reference frame which is non-inertial in the above defined sense.

I guess that's kind of what @vis_insita said?

 

[vis_insita]

Thank you, @Dale, for the friendly welcome.

A reference I intended, but forgot, to include in my first post is Misner, Thorne, Wheeler, Gravitation, which contains a chapter dedicated to the covariant formulation of Newtonian Gravity. This approach goes back to Cartan and is also known as Newton-Cartan-Theory. I am only aware of one other book that also discusses it, but it is in German: Dirschmid, "Tensoren und Felder".

To be honest, I haven't really thought about how the Third Law fits in, since it always appeared somewhat unimportant to me. It may be true or false without affecting much of the rest of the theory as far as I can tell. Because the formulation of the first two laws I have given is completely local, I don't expect that the Third Law can be included in a natural way, since it appears inherently nonlocal to me. (But I am attempting one formulation below.) Still I believe that even in this local formulation the theory is general enough to incorporate all cases of physical interest. As long as the influence of the other particle can be described by a field equation, that has to be solved simultaneously with the equations of motion of the particles I don't see a fundamental problem.

Consider gravity which is a paradigmatic two-body-force obeying the Third Law. Yet, in Newton-Cartan-Theory there is a field equation (very similar to the Einstein equation)

$$(\text{Ricci-Curvature}) = (\text{mass density})\mathrm{d}t\otimes\mathrm{d}t,$$

wich makes the equation of motion of every particle completely local. Of course the fields thus defined will in general not be independent degrees of freedom, since the field equations will not contain time derivatives in the characteristic ways, as is necessary in Relativity. But this doen't invalidate their appearance in Newtonian Mechanics. Also, I think in view of the equivalence principle, gravity is more naturally included among the inertial forces, since it shows up only implicitely in $\nabla$, not in the force law $\boldsymbol{F}$. So here the Third Law used as a criterion for the distinction between real force and fictitious force is somewhat in the way of a covaraint formulation of the equivalence principle.

Now, if one wanted to include non-local interactions, I think one must rephrase the 2nd law. The n-particle force $F_i$ on particle i is not a vector field on space time anymore, but a more complicated object that assigns a vector field to each absolute time t and n-1 events happening at that time. So for every events $x_1, x_2, ...,$ with $t(x_1)=...=t(x_{n-1})$ (all happening at the same absolute time) there is a vector field: $x \mapsto \boldsymbol{F}_i(x, x_1, ...)$ which is tangent to x and spatial. But, since I just made all that up, I'm not sure if it makes sense.

In any case, I believe the Third Law must be regarded as a condition on two body forces which reads "action equals reaction at each point in absolute time". This can be formulated on Newtonian space time as

$$\boldsymbol{F}_1(x(t), y(t)) = -\boldsymbol{F}_2(y(t), x(t))$$

where $x$ and $y$ are the world lines of both particles. This condition makes sense since all vectors are taken at the same absolute time and are thus contained in the same absolute Euclidean space.

 

[vis_insita]

It also clarifies what bothers me... in what space-time does this co-variant derivative exists? It feels like an "arbiter of all Newtonian dilemmas" but I don't get what gives it reference, how does it avoid being accused of being a "preferred frame" or in some sense occupying one, like some background geometry?

A covariant derivative does not define a preferred frame.

The principle of relativity is very elegantly incorporated in the structure of space time. Note that the usual approach to Newtonian Mechanics gives space time a structure $\mathbb{R}\times\mathbb{E}^3$. This somewhat contradicts the principle of relativity since it allows to define absoluet rest as "staying at the same point in absolute space $\mathbb{E}^3$ for all times."

In the covariant formulation one potulates a more general structure: Spacetime is a 4-dimensional manifold $\mathbb{N}^4$, say, on which a scalar field $t: \mathbb{N}^4 \rightarrow \mathbb{R}$ is definied, which assigns each event the absolute time at which it happens. (Of course there are some requirement on $t$, like $\mathrm{d}t\neq 0$ without which the following wouldn't make much sense.) Now you define absolute space at time $\tau$ as the set of events $x$, that satisfy $t(x)=\tau$. The crucial thing is that each time t has its own absolute space. It makes no sense to talk about two events at different times happening in the same place. Thus, you cannot define absolute rest here. The covariant derivative doesn't change anything about that fact. You can easily verify this by showing that for each inertial frame with origin $O(t)$ there is a different one, whose origin $O'(t)$ has a nonvanishing constant relative velocity to the first one, i.e.

$$\nabla_{e_0}(O'(t) -O(t)) \neq 0.$$

I think there is an interesting pattern to recognize here

1) The elementary formulations of Newtonian Mechanics involve a global split of space and time. This makes the Principle of Relativity a pure accident of the dynamics, which only shows up as a symmetry of the force laws that is in no way reflected in the fundamental structure of space time. To remedy this you have to unify space and time, even in Newtonian Mechanics.

2) The elementary formulations of Newtonian mechanics treat gravity as a normal interaction force. This makes the Principle of Equivalence a pure accident of the equality of "inertial" and "gravitational" mass. To adequately represent that principle in the Laws of Motion as well as in the spacetime structure you have to make spacetime curved, even in Newtonian Mechanics.

 

[atyy]

By the way it is natural to include Newtonian Gravity as curvature of space time, by stating the equivalence principle in the form "inertial motion = free fall". Then the Second Law formally holds as stated above. Only $\boldsymbol{F}$ is also free of gravitational forces, which are instead included in $\nabla$ and are in this sense more on par with inertial forces. Also all "inertial frames" become "local inertial frames".

This is important, and shows a great difference between your presentation and the usual Newton's laws that we have been discussing. While the overall physics is the same, the definition of force is not the same in both formalisms.

 

[vis_insita]

This is important, and shows a great difference between your presentation and the usual Newton's laws that we have been discussing. While the overall physics is the same, the definition of force is not the same in both formalisms.

Since I am not using the Third Law anywhere I believe the definition of force is a little more general. But I think this is only a minor difference, not a great one. The point was more that even absent the Thrid Law there is no conceptual problem in formulating the other two laws of motion, or in treating inertial forces, or in defining inertial frames. The Third Law doesn't seem necessary for that.

The treatment of the Equivalence Principle, however, is independent of the treatment of all non-gravitational forces. It is possible to incorporate gravity as spacetime curvature in a covariant formulation, as well as in a non-covariant one. But it is not necessary in either. You can also postulate the existence of a field $\Phi$ on spacetime, which satisfies $\Delta\Phi = \rho$ on each slice of constant absolute time, and couples to the particles via

$$m\nabla_{u}u = -m\nabla\Phi + \boldsymbol{F}.$$

But the gravitational force $-m\nabla\Phi$ is a little elusive and cannot be unambiguously (i.e. observer-independently) assigned to a particular event without further assumptions. Since $\partial_i \Phi$ can always be absorbed in the redefinition

$\tilde{\Gamma}^i_{00} = \Gamma^i_{00} + \partial_i\Phi,$

mechanics provides no basis to distinguish a locally vanishing gravitational field observed by a non-inertial observer with linear acceleration $b^i = \Gamma^i_{00}$ from a non-vanishing field of strength $\Gamma^i_{00} = \partial_i\Phi$ observerd by an inertial oberver.

This is a real physically important difference to other forces and can justifiably influence the way we see and define forces.

 

[atyy]

Since I am not using the Third Law anywhere I believe the definition of force is a little more general. But I think this is only a minor difference, not a great one. The point was more that even absent the Thrid Law there is no conceptual problem in formulating the other two laws of motion, or in treating inertial forces, or in defining inertial frames. The Third Law doesn't seem necessary for that.

The treatment of the Equivalence Principle, however, is independent of the treatment of all non-gravitational forces. It is possible to incorporate gravity as spacetime curvature in a covariant formulation, as well as in a non-covariant one. But it is not necessary in either. You can also postulate the existence of a field $\Phi$ on spacetime, which satisfies $\Delta\Phi = \rho$ on each slice of constant absolute time, and couples to the particles via

$$m\nabla_{u}u = -m\nabla\Phi + \boldsymbol{F}.$$

But the gravitational force $-m\nabla\Phi$ is a little elusive and cannot be unambiguously (i.e. observer-independently) assigned to a particular event without further assumptions. Since $\partial_i \Phi$ can always be absorbed in the redefinition

$\tilde{\Gamma}^i_{00} = \Gamma^i_{00} + \partial_i\Phi,$

mechanics provides no basis to distinguish a locally vanishing gravitational field observed by a non-inertial observer with linear acceleration $b^i = \Gamma^i_{00}$ from a non-vanishing field of strength $\Gamma^i_{00} = \partial_i\Phi$ observerd by an inertial oberver.

This is a real physically important difference to other forces and can justifiably influence the way we see and define forces.

Would it be correct to think that what you have added beyond the traditional Newton's first and second, is that you have specified their transformation properties in arbitrary coordinates?

 

[vanhees71]

That's not the point. The range of interactions doesn't matter if forces would need no interactions at all. How do you exclude that without Newton III?

Now I'm completely lost. If the forces are not interactions, Newton III doesn't make any sense.

It's of course clear that a physical theory is not simply a system of axioms as in mathematics but you have to operationally define it. Again, I can only repeat the logic of Newton's arguments:

Newton I: Assumes the existence of an absolute time and space, where the law of inertia holds, i.e., bodies which are not "influenced" by other bodies (and for Newton there's nothing else than material bodies of course, because fields haven't been discovered yet) move in rectilinear uniform motion in such an "inertial frame".

Newton II: Forces acting on a body change the state of motion within an inertial frame, being proportional to the acceleration of a body. The proportionality constant is a measure for the inertia of the body and also a meausure for "amount of substance".

Newton III: Forces act pair-wise and if $\vec{F}_{12}$ is the force on body 1 through interaction with body 2, the force on body 2 through interaction with body 1 is $\vec{F}_{21}=-\vec{F}_{12}$.

As we know today, that's not even complete since also within Newtonian physics there are generic $N$-body forces with $N>2$ (e.g., the nucleon-nucleon interaction in relativistic models).

It's anyway a very complicated issue, and from a modern point of view it's simpler to just state the space-time model and analyze its symmetries, but that we've also discussed already above.

 

[vis_insita]

Would it be correct to think that what you have added beyond the traditional Newton's first and second, is that you have specified their transformation properties in arbitrary coordinates?

I would say that the transformation properties to arbitrary coordinates are made especially transparent in this approach. But really the only thing that is added beyond traditional approaches is, I think, clarity. It is perfectly valid to base Newtonian Mechanics on the notion of inertial frames and derive everything, including the form of the equations of motion, in non-inertial frames or arbitrary coordinate systems from that.

But personally I always found that a little confusing. And I encountered many statements that I suspect are the result of similar confusions. Some such statements are plainly false, e.g. the very common claim that Newton's axioms are only valid in inertial frames. Against such claims a generally covariant formulation of those axioms should provide a decisive argument.

One of the subtleties that can in the traditional approach become the more confusing the more you think about it, is the derivation of inertial forces. It usually involves a discussion of the time-dependence of basis vectors which goes like this:

In an inertial frame S we have
$$\frac{d v}{d t} = \dot{v}^i e_i \qquad (1).$$
But in a non-inertial frame S' the basis vectors move, so we have to take that into account too. Thus
$$\frac{d v}{d t} = \dot{v}^ie'_i + v^i\dot{e}'_i.$$
But what justifies the asymmetry? The basis vectors of S' clearly appear to move if viewed from S. But so do the basis vectors of S appear to move when viewed from S'. I once had a discussion where I pointed out that condition (1) is precisely one of the characteristics of an inertial frame. My opponent replied that this cannot be true, since from the point of view of an observer at rest in a rotating frame, the basis vectors don't move. So for that observer the same condition holds. I said that the time derivative of a vector is again a vector, which is a geometric object. So it either is or isn't zero. But, in any case, that statement is independent of any reference frames. I believe he remained unconvinced, but I wonder how he would derive inertial forces.

I think the cause for this confusion is that an important part of mathematical structure, namely the covariant derivative, is completely absent from the discussion. It is never really made clear what the inertial motion of particles has to do with time derivatives of basis vectors. The covariant approach makes that perfectly clear: $\nabla$ defines straightness, i.e. inertial motion, as well as vector derivatives.

 

[atyy]

I would say that the transformation properties to arbitrary coordinates are made especially transparent in this approach. But really the only thing that is added beyond traditional approaches is, I think, clarity. It is perfectly valid to base Newtonian Mechanics on the notion of inertial frames and derive everything, including the form of the equations of motion, in non-inertial frames or arbitrary coordinate systems from that.

In practice the covariant first law is still useless if there are no non-interacting particles, so that seems the same as the elementary version.

The covariant second law is still a force definition, as in the elementary version, since it is empty unless F is specified.

The covariant second law seems able to distinguish non-inertial forces from real forces, but similarly to the elementary version, you add something. In the elementary version, one of the things one can add is Newton's 3rd law, but that doesn't have to be the extra ingredient, it can also be the transformation properties of the forces.

However, in the elementary version, adding transformation properties does not distinguish between an accelerated frame and a uniform gravitational field. Thus an accelerated frame can cancel a uniform gravitational field not only locally, but over a large region of space and time. Does the covariant version have any advantage in distinguishing an accelerated frame from a uniform gravitational field?

 

[vis_insita]

In practice the covariant first law is still useless if there are no non-interacting particles, so that seems the same as the elementary version.

The covariant second law is still a force definition, as in the elementary version, since it is empty unless F is specified.

It is true that both the covariant and the non-covariant version have the same content. This is so by intention. But we have to be clear about what that content is.

The First Law both defines what "straight uniform motion" means, and states a physical condition under which it occurs. The latter condition would also follow from the Second Law. But the definition of "straight uniform motion" is not useless in the sense that it could be discarded without substitution. From a theoretical point of view I think it is most important.

Also it seems to me that traditional expositions are usually not very consistent in applying that definition even if they carefully formulate it. E.g. a non-covariant formulation that would be (absent gravitation) equivalent to the one I have given is

1st Law: "There exists a frame of reference in which all straight uniform trajectories have the form
$$\left(\frac{\mathrm{d}^2 x^i}{\mathrm{d} t^2}\right)\boldsymbol{e}_i = 0. \qquad (1)$$
Isolated, non-interacting particles follow those straight uniform trajectories."

Now, changing to an arbitrary reference frame will in general violate condition (1) in a well-defined way, but it will not change anything about the straightness or the uniformity of the motion, because both properties have absolute, frame-independent meaning within the theory. One cannot neglect defining them in a frame-independent way without confusing everything. Yet, some authors seem to say that condition (1) defines straightness in any frame which absolutely makes no sense. Thus inertial forces are introduced to "explain" the departure from straight uniform motion in light of the Second Law, which is completely backwards. Inertial forces occur even if the motion is straight and uniform merely as a consequence of chosing a non-inertial reference frame.

The covariant second law seems able to distinguish non-inertial forces from real forces, but similarly to the elementary version, you add something. In the elementary version, one of the things one can add is Newton's 3rd law, but that doesn't have to be the extra ingredient, it can also be the transformation properties of the forces.

I think here you are saying that inertial forces arise as a consequence of the transformation properties of forces. They do not; at least not in the way they are usually introduced in textbooks. They arise because the time derivative that defines acceleration is caluclated in a time-dependent basis. Thus, the only transformation property involved is that of the time derivative
$$\left(\frac{\mathrm{d}}{\mathrm{d}t}\right)_{S}[\cdot] = \left(\frac{\mathrm{d}}{\mathrm{d}t}\right)_{S'}[\cdot] + \boldsymbol{\omega}\times[\cdot]$$
which is of fundamental importance to the theory and is, as the covariant formulation makes clear, nothing but a special case of the usual "transformation law" of an affine connection.

(Maybe this equation is that something you say must be "added" to make a clear distinction between real and fictitious forces. I think I would agree with that.)

However, in the elementary version, adding transformation properties does not distinguish between an accelerated frame and a uniform gravitational field. Thus an accelerated frame can cancel a uniform gravitational field not only locally, but over a large region of space and time. Does the covariant version have any advantage in distinguishing an accelerated frame from a uniform gravitational field?

Why would that be an advantage? I think it is precisely the content of the Equivalence Principle that you cannot distinguish both situations. In the Newton-Cartan-Theory both the uniform acceleration and the "local gravitational field" originate from
$$\nabla_{\boldsymbol{e}_0}\boldsymbol{e}_0 = \Gamma^i_{00}\boldsymbol{e}_i$$
As in General Relativity that term can always be made to vanish at the origin, by changing to a local inertial frame. If there is only a uniform "gravitational force" (which really isn't a gravitational force at all) it vanishes everywhere and the local inertial frame is a global one.

 

[Dale]

There exists a frame of reference in which all straight uniform trajectories

Would it be better to just say “geodesic” here?

 

[vis_insita]

Would it be better to just say “geodesic” here?

In my opinion this is exactly the meaning that should be attributed to "straight uniform motion" also in Newtonian Mechanics, yes.

 

[DrStupid]

If the forces are not interactions, Newton III doesn't make any sense.

I'm not sure what you mean. Newton III defines forces to be interactive. Thus, I would reather say that non-interactive forces make no sense (due to Newton III).

Again, I can only repeat the logic of Newton's arguments:

As we can't ask Newton anymore, nobody knows for certain what the logic of his arguments really is. You just repeat your opinion about them. Here is my opinion, based on the original wordings^1,2:

Newton’s laws of motion are about forces. That’s why it makes sense to start with Newton’s definition of force:

"Def. IV: Vis impressa est actio in corpus exercita, ad mutandum ejus sta-tum vel quiescendi vel movendi uniformiter in directum."
(An impressed force is an action exerted upon a body, in order to change its state, either of rest, or of moving uniformly forward in a right line.)

With this definition we have a general impression what forces are (something that changes the state of motion of bodies) and what they not are (something that preserves the state of motion [e.g. in contrast to the Aristotelian concept of force]). But it is not ready to use. There are some questions open:

1. The definition of force says that it is an action that changes the state of motion of a body, but are there other reasons for a body to change its motion, in particular, can it spontaneously change its motion without an action exerted upon it? That is answered by

"Lex I: Corpus omne perseverare in statu suo quiescendi vel movendi uniformiter in directum, nisi quatenus a viribus impressis cogitur statum illum mutare."
(Law I. Every body perseveres in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed thereon.)

That means that forces are the only reason for a change of the state motion.
Now we know that every force changes the state of motion of a body (according to Def. IV) and every change of motion is caused by a force (according to Law I). Or

$F = 0 \Leftrightarrow a = 0$

which is logical equivalent to

$F \ne 0 \Leftrightarrow a \ne 0$

2. Now we have a full qualitative relation between forces and changes in the state of motion but what how to quantify it? That is answered by

"Lex II: Mutationem motus proportionalem esse vi motrici impressae, & fieri secundum lineam rectam qua vis illa imprimitur."
(Law II: The alteration of motion is ever proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed.)

In modern notation and with a unified system of units this means

$F = \dot p$

3. Together with the definition of momentum (Def. II) we now have a full qualitative and quantitative relation between forces and changes in the state of motion. But where do forces come from? That is answered by

"Lex III: Actioni contrariam semper & aequalem esse reactionem: sive corporum duorum actiones in se mutuo semper esse aequales & in partes contrarias dirigi."
(Law III: To every action there is always opposed an equal reaction: or the mutual actions of two bodies upon each other are always equal, and directed to contrary parts.)

That implies (but is not limited to)

$action \Rightarrow reaction$

which is logical equivalent to

$\neg reaction \Rightarrow \neg action$

Forces are limited to interactions. Without interaction there is no force.

Law III says furthermore that the interactions need to be symmetrical. For two bodies A and B the force $F_{AB}$ exerted by A upon B and the counter-force $F_{BA}$ exerted from B upon A must comply with

$F_{AB} = - F_{BA}$

That also implies that the interactions are binary only. For more than two bodies the additivity of forces (resulting from the additivity of momentum and also described in Corollary I) would for infinite different linear combinations. Newton III tells us that the combination of symmetric pairs of forces shall be used only.

That’s what Newton’s laws of motion say to my understanding.

What has that to do with inertial frames? The laws of motion (with the wording above) are not valid in all frames of reference. In #85 I demonstrated how rotating frames violate them. They can either comply with Law I and II or with Law III but not with all of them at once. Frames of references that comply with all laws of motion are called inertial. Of course there are other possibilities to define inertial frames, but that’s how it works with the laws of motion as given above.

If and how the laws of motion define inertial frames strongly depends on the exact wordings. There are a lot of different versions out there, resulting in different conclusions. In #34 I already mentioned a version that has been introduces by Newton himself. By deleting the first part of Newton III ("To every action there is always opposed an equal reaction") he allowed fictitious forces and therefore extended the laws of motion to non-inertial frames. There are a lot of cases where something like this is very useful. But with this modification the laws of motion alone do not define inertial frames anymore.

That’s why it is very important to clarify what we actually mean with “Newton’s laws of motion”. I refer to the original or equivalent wordings. If you refer to another version that please post the wording.¹Latin original: http://cudl.lib.cam.ac.uk/view/PR-ADV-B-00039-00001/1
² Eglish translation: https://en.wikisource.org/wiki/The_Mathematical_Principles_of_Natural_Philosophy_(1846)