# Foundations: Newton's Third Law and time reversal invariance

Gold Member

## Main Question or Discussion Point

Let me propose a list of principles of classical dynamics, specifically designed for education, for introduction to novices:

- In the absence of any force: objects in motion move along straight lines, covering equal distances in equal intervals of time
- Composition of motion: position vectors, velocity vectors and acceleration vectors add according to the rules of vector addition in Euclidean space
- Time reversal invariance
- F=m*a

I'm not claiming that the above list hits the nail on the head, but I do think it's an improvement over Newton's formulation of the Three laws. We learned a lot over the past centuries, and we are in a position to improve on how the principles are formulated.

I believe that the third item, time reversal invariance, is an absolute must. Nöther's theorem links symmetries in physics theories to conserved quantities; in newtonian dynamics time reversal invariance is correlated with energy conservation.

In the Principia Newton asserts the Third Law as a blanket approach to statics and dynamics. I am highly skeptical about that conflation of statics and dynamics. From here on, when I refer to the third law, I mean exclusively its application in dynamics.

Perfectly elastic collision is time reversal invariant, as a matter of principle. I take it as obvious that in a Universe in which the third law doesn't exist collisions won't be time reversal invariant. Does that also work the other way round? In a Universe with time reversal invariant laws of motion, does it follow logically that the third law must hold good? I think so, but I'm interested in comments.

While newtonian physics is exuberantly alive, I believe that Newton's original formulation of the Three Laws is badly outdated. In retrospect we see that Newton's First law and Third Law are assertions of symmetry principles. Those principles are more profoundly expressed as symmetries of space and time. Novices are confused by the Third law because the Third law is so awkward.

Cleonis

larsa

## Answers and Replies

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D H
Staff Emeritus
Science Advisor
I believe that the third item, time reversal invariance, is an absolute must.
Why is that an absolute must?

Time reversibility is not nearly as important concept for elementary physics education as are the conservation laws. In particular, time reversibility and Newton's third law are not related. Newton's third law follows from conservation of momentum (linear and angular), not from time reversibility. Time reversibility implies that mechanical energy is conserved.

To truly understand Noether's theorem one must have an understanding of Lagrangian mechanics. The mathematics of that formulation is beyond the reach of high school students and college freshman. A force-based formulation is well within the grasp of someone who knows introductory calculus.

Finally, physics education does not teach how Newton envisioned things. Vector notation is a relatively recent development, a bit over 100 years old. Newton used highly geometric arguments and archaic notation.

Gold Member
Newton's third law follows from conservation of momentum (linear and angular), not from time reversibility.
My criticism is that textbooks keeps quoting the third law as a separate principle, which is awkward.
I gather that you go along with that particular aspect of what I stated; you regard the third law as following from first principles.

Teachers should stop bugging novices with the third law, and work with conservation of momentum.

Conservation of momentum follows from the First and Second law:
If two objects exert a force upon each other, accelerationg each other, then their common center of mass will keep moving in a straight line, covering equal distances in equal intervals of time.

As to energy, I think you have a point; introduction of the energy concept can be deferred until later.

Cleonis

larsa
D H
Staff Emeritus
Science Advisor
Conservation of momentum follows from the First and Second law
Conservation of linear momentum for a single particle *is* the first law. For a system of particles, conservation of momentum does *not* follow from the first two laws. You need the third law to derive conservation of linear momentum.

Gold Member
Newton's third law follows from conservation of momentum
You need the third law to derive conservation of linear momentum.

You seem to be going round in a circle.

Seriously now: my assessment is that conservation of momentum is more profound.

Let two objects, A and B, each with known inertial mass, be moving through space, and they exert a force upon each other (let's say they are connected by a stretched bungee cord.)

Then you find from the second law how much force each one can exert. Let the mass of A be twice the mass of B. Then it follows from the second law that B will accelerate twice as hard as A.

My point is, A and B are floating in space, and you can't brace yourself in space. The amount of force that B can exert on A is a function of B's inertial mass and B's acceleration, as codified by the second law: F=m*a , and the same goes for A.

It follows from the Second law that the two objects exerting a force upon each other cannot cause acceleration of their common center of mass. So why declare an additional principle there, it's already covered.

Cleonis

D H
Staff Emeritus
Science Advisor
You seem to be going round in a circle.
Only when you quote me out of context. :grumpy:

Doc Al
Mentor
Then you find from the second law how much force each one can exert. Let the mass of A be twice the mass of B. Then it follows from the second law that B will accelerate twice as hard as A.
Only if the force on each is the same, which follows from the third law.

Gold Member
Then you find from the second law how much force each one can exert. Let the mass of A be twice the mass of B. Then it follows from the second law that B will accelerate twice as hard as A.
Only if the force on each is the same, which follows from the third law.
Let's take a closer look at that.

In the previous version of the illustrating example I mentioned a bungee cord. I replace that now with a cable that can be reeled in.

Two spaceships, A and B, among a larger formation of spaceships floating around, are connected by a cable, initially not under tension. For simplicity the cable is thought of as having negligable mass.

Scenario 1: A reels in the cable
Scenario 2: B reels in the cable
Scenario 3: A and B both reel in the cable.

You are observing from a distance and the only information you can gather is A's and B's respective acceleration with respect to the common center of mass of the entire formation of floating spaceships.

Now, is there any scenario, any observation that you could make, that would lead you to the conclusion: "Hey, A is exerting a force on B, but B is not exerting a force on A."
What would have to happen to force just that conclusion?

I don't think any such scenario is available. That is why I claim that proposing the Third law as an independent principle is redundant.

Cleonis

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Doc Al
Mentor
Now, is there any scenario, any observation that you could make, that would lead you to the conclusion: "Hey, A is exerting a force on B, but B is not exerting a force on A."
What would have to happen to force just that conclusion?
View things from a frame in which A and B are at rest before they start pulling. B pulls A. A accelerates; B does not.

Of course we don't expect that since we know Newton's third law.

I don't think any such scenario is available. That is why I claim that proposing the Third law as an independent principle is redundant.
You're going in circles. The reason why you can't find such a scenario is that Newton's third law holds.

Gold Member
View things from a frame in which A and B are at rest before they start pulling. B pulls A. A accelerates; B does not.

Of course we don't expect that since we know Newton's third law.
Let's take a closer look at that.

If you want a scenario in which no acceleration is attributed to B, then you must map the motion in a coordinate system that is all the time co-moving with B. As we know, that is an accelerating coordinate system.

As we know, the laws of motion hold good only for the equivalence class of inertial coordinate systems.

Your reasoning above does not establish necessity for the Third law. In order to have the whole set of laws of motion in the first place the motion must be referenced to an inertial coordinate system.

Cleonis

Doc Al
Mentor
If you want a scenario in which no acceleration is attributed to B, then you must map the motion in a coordinate system that is all the time co-moving with B. As we know, that is an accelerating coordinate system.
If no force acts on B, then B is at rest in an inertial frame. (Newton's first law.)

Gold Member
If no force acts on B, then B is at rest in an inertial frame. (Newton's first law.)
OK, there's been a Babylonian misunderstanding.

Going back to the example:
Two spaceships, A and B, among a larger formation of spaceships floating around, are connected by a cable, initially not under tension. For simplicity the cable is thought of as having negligable mass. The cable is reeled in.

Let's say that the following is observed: as the cable is reeled in: B doesn't move, but A does, towards B.

What we agree on, I assume, is that that scenario cannot happen if the third law is in place.

What I argue is that you already get a violation of physics laws even outside consideration of the third law. For A to start moving (towards B) a force must be exerted upon A. What is the source of the force upon A supposed to be? Without a source for that force you get a violation of the concept of cause and effect.

That raises the question: is it necessary to assume the existence of causality? I argue it is, I think it's a foundation of classical dynamics that causality exists.

Well, I started this thread to invite comments.
I find that under the influence of the comments from you and D.H. my thoughts are changing. I started the thread feeling the third law is redundant, but in retrospect I was shaky on why that is so. Initially I thought about time reversal invariance. I have shifted, and by now I have arrived at thinking that if the existence of causality is granted then the third law is redundant.

Cleonis

Dale
Mentor
I believe that the third item, time reversal invariance, is an absolute must. Nöther's theorem links symmetries in physics theories to conserved quantities; in newtonian dynamics time reversal invariance is correlated with energy conservation.
Actually, it is time translation invariance, not time reversal invariance. Noether's theorem applies to differentiable symmetries, i.e. symmetries that can be made in infinitesimal amounts. Time translation symmetry is a differentiable symmetry, but not time reversal symmetry.

That said, I agree. The Lagrangian formulation of classical mechanics should be introduced as early as possible, probably immediately after the introduction of energy.

Gold Member
Actually, it is time translation invariance, not time reversal invariance. Noether's theorem applies to differentiable symmetries, i.e. symmetries that can be made in infinitesimal amounts. Time translation symmetry is a differentiable symmetry, but not time reversal symmetry.
Thank you for setting me straight on that one. I took the subject too lightly.

[...] The Lagrangian formulation of classical mechanics should be introduced as early as possible [...]
Actually, I did not intend to make any statement about The lagrangian formulation, and I don't have a opionion about that one way or the other.

Cleonis

Dale
Mentor
Actually, I did not intend to make any statement about The lagrangian formulation, and I don't have a opionion about that one way or the other.
Oh, sorry, I misunderstood your point.

Out of curiosity, how would you introduce Noether's theorem and the idea of conserved quantities and symmetries without introducing Lagrangian mechanics? They always seemed fairly inseparable to me.

Gold Member
Out of curiosity, how would you introduce Noether's theorem and the idea of conserved quantities and symmetries without introducing Lagrangian mechanics? They always seemed fairly inseparable to me.
I'm not professionally involved in education, but Nöther's theorem is by nature a very advanced subject, so I would definitely decide against introducing/mentioning that to novices.

I was casting around for a principle that would allow me to demote Newton's third law to the status of theorem, instead of axiom. In retrospect time reversal invariance wasn't a good candidate for that.

Time reversal invariance is an important, deep issue in classical mechanics, but right now I'm unsure whether to think of it as a theorem or axiom.

Any derivation of conservation of angular momentum has as ingredient that space is the same in all directions; rotational symmetry of space. Likewise I have an intuitive notion of momentum being correlated with symmetry of space. Intuitively, energy conservation being correlated with a symmetry of time slots in well with that. The Lagrangian formulation allows formalized statement of these correlations, but an intuitive grasp is possible, I think. Other than that, like I said earlier I was taking the subtleties of Nöther's theorem too lightly, I'll be more cautious in the future.

Cleonis

I always thought the first law was redundant, because it was a trivial case of the second law: if there is no net force on an object, its acceleration is zero and it moves with constant velocity. Is there any new information contained in the first law that is not contained in the second???

Gold Member
I always thought the first law was redundant, because it was a trivial case of the second law: if there is no net force on an object, its acceleration is zero and it moves with constant velocity. Is there any new information contained in the first law that is not contained in the second???
The First law can be reformulated as follows:
(Arguably, it must be so reformulated.)
"Space is uniform and time flows uniformly, such that if an object is not subject to any force it will move in a straight line, covering equal distances in equal intervals of time."

Stated in that way the first law asserts that space and time are Euclidean in nature.
In Newton's time asserting that would have appeared perfectly superfluous, but according to the general theory of relativity space and time are not euclidean, so today we know it's by no means trivial.

We cannot know Newton's considerations at the time, but I like to think that Newton did perceive that asserting the uniformity of space and time is so important that it's worth elevating it to the status of axiom.

Kepler's law of areas is an example of a principle that correlates uniform flow of time with a symmetry of space: equal areas are swept out in equal intervals of time. The First law can be regarded as laying the foundation for that, in correlating a symmetry of space with uniform flow of time. "If an object is not subject to any force it will move in a straight line, covering equal distances in equal intervals of time."

Cleonis

vanesch
Staff Emeritus
Science Advisor
Gold Member
In fact, the first law comes down to:

there exists at least one inertial frame.

After all, you first need to postulate the existence of such a frame before being able to even write down the second law (F = m a).

From the second law then follows that if there exists ONE inertial frame, there exists a whole family of inertial frames (essentially the Galilean group).

The third law is a property of forces, which doesn't need to follow just from the first and the second law. After all, Noether's theorem only works if a Lagrangian description exists, but there doesn't need to exist a Lagrangian description for all force systems compatible with the first and the second law of Newton. In other words, the set of possible dynamics that satisfies the first and the second law is much larger than the set of dynamics that have a Lagrangian description, in which one could use Noether's theorem. It turns out that nature has its dynamics entirely in this second, more restricted set, but that didn't need to be the case.

It could be fun to think of dynamical systems that respect the first and second law, but not the third one (I think you will end up with cartoon physics in a way - like pulling yourself up by your hair and so on: all perfectly compatible with the first and second law).

larsa
Newton's Third Law

Newton's third law of motion is a very important piece of work in physics, but not taught correctly in ecademics. Let me explain this law in bit more elaborate fashion:

1) Inside an accelerated spaceship

When inside an accelerated spaceship, you experience your weight opposite to the direction of acceleration.

Your weight is proportional to acceleration and the direction is opposite.

2) On the surface of earth

If the direction of acceleration due to earth's gravity is towards its centre, why do we feel our weight in the same direction, contrary to the first finding?

Blame it on schoold text books that have taught us about gravity incorrectly.
Imagine you are inside a freely falling elevator, you feel no weight and your acceleration with respect to the elevator is 0 (zero). Elevator is your reference frame that is falling towards the centre of earth not because of earth's gravity but because of curved space-time around it. In that curved space-time when you are moving towards earth's centre with acceleration 9.8 m/s2, you are actually not accelerating with respect to your inertial reference frame, and you feel no weight.

When you hit the surface of earth, your acceleration with respect to the centre of earth is 0 zero, however your acceleration with respect to your reference frame is -9.8 m/s2. The direction of this acceleration is away from the centre of earth and you feel your weight towards the centre of earth. This explanation seems to fix the problem.

Your weight again is proportional to acceleration and the direction is opposite.

3) On a rotaing planet

Rotation causes acceleration towards the centre of the body. Curved space-time causes acceleration away from the centre of a planet as we have seen above. Because their directions are opposite, the net acceleration reduces. On a rotating planet you feel less of weight, on a fast rotating planet, your weight may become zero. And on a very fast rotating planet you would feel your weight away from the centre of that planet and not towards the centre and a rope would be required to keep you from spinning off that planet.

In all three scenarios we have seen that a body's weight is proportional to acceleration and the direction is opposite. Now let me remove unscientific terms like weight, centrifugal force, centripetal force etc. These terms are more of jargons rather than scientific.

Thus your gravitaional mass is proportional to acceleration and the direction is opposite. This I believe is the correct representation of Newton's Third law of motion, and this cannot be removed from the books of physics, it is very important!

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vanesch
Staff Emeritus
Science Advisor
Gold Member

Newton's third law of motion is a very important piece of work in physics, but not taught correctly in ecademics.

When inside an accelerated spaceship, you experience your weight opposite to the direction of acceleration.
I don't know what "academics" you are referring to, but this is normally taught everywhere (d'Alembert's force): in an accelerated reference frame, you undergo "forces" (inertial forces or fictious forces) that are equal to - a m (a = acceleration of the point in the accelerated reference frame wrt an inertial frame and m is the mass of the object being there).

Blame it on schoold text books that have taught us about gravity incorrectly.
Imagine you are inside a freely falling elevator, you feel no weight and your acceleration with respect to the elevator is 0 (zero). Elevator is your reference frame that is falling towards the centre of earth not because of earth's gravity but because of curved space-time around it. In that curved space-time when you are moving towards earth's centre with acceleration 9.8 m/s2, you are actually not accelerating with respect to your inertial reference frame, and you feel no weight.

When you hit the surface of earth, your acceleration with respect to the centre of earth is 0 zero, however your acceleration with respect to your reference frame is -9.8 m/s2. The direction of this acceleration is away from the centre of earth and you feel your weight towards the centre of earth. This explanation seems to fixe the problem.
I really don't know what "school books" you are referring to, because this is also the standard explanation.

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Doc Al
Mentor

So your gravitaional mass is proportional to acceleration and the direction is opposite. This I believe is the correct representation of Newton's Third law of motion, and this cannot be removed from the books of physics, it is very important!
It's unclear from your post that you even know what Newton's Third law is.

vanesch!

If you followed the thread correctly, you would know its the primary schools kids we are talking about. In primary schools gravity is introduced as a force with direction of acceleration towards the centre of earth. This explanation of gravity will not help kids understand the Third Law.

I do accept that it would be better to use the term weight rather than gravitational mass for school kids easier comprehension.

As far as the below terms are concerned

d'Alembert's force
Fictitious force
pseudo force
centripetal force
centrifugal force

They are just colorful jargons, they don't exist and would only make physics seem more complicated for kids. The only forces that exist in nature are:

Gravitational force
Electromagnetic force
Weak nuclear force
Strong nuclear force

Do you get it love?

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Doc Al
Mentor

If you followed the thread correctly, you would know its the primary schools kids we are talking about. In primary schools gravity is introduced as a force with direction of acceleration towards the centre of earth. This explanation of gravity will not help kids understand the Third Law.
In physics 101 (pre-general relativity), the weight of an object is treated as a force (gravity) exerted by the earth and directed towards its center. Where do you see a problem with this formulation and the Third Law?

Please describe what you think the Third Law says.

My dear Doc Al!

It's unclear from your post that you even know what Newton's Third law is.
Do you understand the Third Law?

Your explanation should be like, "when you push the wall the wall pushes you back with the same force", shouldn't it be?

Reciprocal action means 'increase in your weight in the opposite direction'. So when you apply a force on a body, the body's weight increases in the opposite direction as explained in the above three examples.

"When you push the wall the wall pushes you back with the same force"! What an explanation!

Good night!