# Inertia and action-reaction

• Ranku

#### Ranku

Newton’s third law, to every action there is an equal and opposite reaction, is valid in various situations for various reasons. If an object is pushing against another object, the other object reacts with an equal and opposite force. At the same time, inertial force is also arising upon the second object that is equal and opposite to the applied force upon the object. Is the inertial force arising upon the second object also responsible for the object reacting with equal and opposite force to the first object?

There is no "inertia force." If you think it exists, look for its reaction which is no where to be found. This is a persistent misunderstanding.

There is no "inertia force." If you think it exists, look for its reaction which is no where to be found. This is a persistent misunderstanding.
When an object is accelerated due to an applied force, it does experience inertia, doesn’t it?

"Inertia" and "inertia force" are not the same thing.

Look at a typical statement of Newton's Second Law (for a particle): The sum of ALL forces acting on the particle is equal to the product of mass and acceleration.

Note that ALL the (real) forces are included in the first part, "the sum of all the forces." There is no mention of an "inertia force" because there is no real force due to inertia.

When an object is accelerated due to an applied force, it does experience inertia, doesn’t it?
I don’t know what it means to “experience inertia”.

• vanhees71
When an object is accelerated due to an applied force, it does experience inertia, doesn’t it?
It has inertia, which is quantified by its mass. If you mean inertial forces that arise in non-inertial reference frame, those do not obey Newtons 3rd Law.

• vanhees71
Suppose I have a block resting on a horizontal frictionless surface. If I push on it with my hand in the horizontal direction, it will accelerate in that direction according to Newton's laws. As I do this, the block exerts an equal and opposite force on my hand.

Are you calling one of these forces the "inertial" force or are you saying that there is a third horizontal force that I omitted? If so, you should be able to point at some specific agent external to the block that exerts such a horizontal force. Of course there is the Earth and the frictionless surface but these exert vertical forces only so they don't count.

It has inertia, which is quantified by its mass. If you mean inertial forces that arise in non-inertial reference frame, those do not obey Newtons 3rd Law.
So what is the primary distinction between inertial force that arises in a non-inertial reference frame and reaction force in an action-reaction pair?

So what is the primary distinction between inertial force that arises in a non-inertial reference frame and reaction force in an action-reaction pair?
That the inertial force in a non-inertial reference frame is not part of a Newton's 3rd Law pair.

• vanhees71 and Ranku
Suppose I have a block resting on a horizontal frictionless surface. If I push on it with my hand in the horizontal direction, it will accelerate in that direction according to Newton's laws. As I do this, the block exerts an equal and opposite force on my hand.

Are you calling one of these forces the "inertial" force or are you saying that there is a third horizontal force that I omitted? If so, you should be able to point at some specific agent external to the block that exerts such a horizontal force. Of course there is the Earth and the frictionless surface but these exert vertical forces only so they don't count.
I mean inertial force that arises in an object due to acceleration of the object, i.e., due to the object being in a non-inertial frame of reference.

That the inertial force in a non-inertial reference frame is not part of a Newton's 3rd Law pair.
Is there a physical explanation to Newton’s 3rd Law - why is there a reaction to action?

Is there a physical explanation to Newton’s 3rd Law - why is there a reaction to action?
Yes. Newton’s third law is the conservation of momentum. This in turn is due to a symmetry of the laws of physics.

• sophiecentaur and vanhees71
Yes. Newton’s third law is the conservation of momentum. This in turn is due to a symmetry of the laws of physics.
Interesting. Could you maybe elaborate on ”symmetry of the laws of physics” and how that leads to conservation of momentum?

Interesting. Could you maybe elaborate on ”symmetry of the laws of physics” and how that leads to conservation of momentum?
That is known as Noether’s theorem. It basically shows that every conserved quantity is associated with a symmetry in the laws of physics (and vice versa). Noether’s theorem is probably one of the most important and influential concepts in all of physics.

Conservation of momentum goes with the spatial translation symmetry. Basically, the laws of physics are the same here and on Mars and in Andromeda. That symmetry means that momentum is conserved, and therefore we get Newton’s 3rd law.

• Ranku
Off the top of my head, Noether's theorem does not explain conservation of momentum. It just shows how the conservation of momentum is reflected on the symmetry properties of the Lagrangian that describes the dynamics of the system. We don't know why momentum is conserved. It just happens universally and so we make sure to incorporate it in our best theories.

We can show that under certain conditions (e.g., no particle momentum is lost to momentum of electromagnetic fields), conservation of momentum is equivalent to action-reaction. That again that does not prove action-reaction. We just trade one mystery for another. We have no clue whatsoever why when two balls collide head-on, the ratio of their accelerations is a constant which is exactly what action-reaction says (force/cause is not a primitive or necessary concept in physics).

Similarly, we have no clue why two forces (defined as F=ma) add like vectors, no clue why inertial reference systems, where the laws of physics take their simplest form, exist, no clue why the twin "paradox" takes place. (We can only show that it is mathematically equivalent to the assumption that the speed of light is constant but we don't know why the speed of light is constant.). Science does not explain anything. It only provides logically consistent, economical descriptions of our sense data that lead to predictions and applications.

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• Dale and weirdoguy
Noether's theorem tells us that momentum conservation follows from spatial translation invariance, which is a symmetry of Newtonian as well as special-relativistic (Minkowski) spacetime. Thus the dynamical laws on the fundamental level must obey this symmetry and that's "why" the total momentum of a closed system is conserved. Indeed, arguing in this way, the symmetries of the spacetime model determine how the dynamical laws look like. This approach has been very successful in discovering the Standard Model of elementary particle physics.

• Dale and LittleSchwinger
I don’t know what it means to “experience inertia”.
Try teaching.

• • Hornbein, strangerep, vanhees71 and 1 other person
So what is the primary distinction between inertial force that arises in a non-inertial reference frame and reaction force in an action-reaction pair?
What AT says. A nononertial force on an object is not due to some fundamental interaction between the object and something else; it's due to an interaction which causes a force acting on you. But you, being accelerated, attribute a force on the object. There is however no interaction that causes this inertial force, other than on you.

Noether's theorem does not explain conservation of momentum. It just shows how the conservation of momentum is reflected on the symmetry properties of the Lagrangian
And that is a good explanation. An explanation takes something that is complicated or objectionable and reduces it to something that is simple or accepted.

In this case, if the laws of physics are the same here as they are there then Noether’s theorem shows us that momentum is conserved. Our experience learning how to move our bodies and successfully perform physical tasks indicates that the laws of physics are the same here and there. So the symmetry is generally acceptable.

Indeed, if the laws of physics were not the same here as there then we would want an explanation for that and would be dissatisfied without an explanation. The symmetry is a default, and in fact other symmetries are usually taken as the default too, not needing further explanation.

Science does not explain anything. It only provides logically consistent, economical descriptions of our sense data that lead to predictions and applications.
I think that you are putting too much burden on the word “explain” such that the word is meaningless. The way you use the word, it seems like “explanations” are simply not possible.

In science we do something more than simply provide “descriptions of our sense data”. We also abstract our data. We notice for several specific pairs of balls colliding that the ratio of their accelerations is a constant, from that we abstract the idea of a conserved quantity we call momentum. Momentum is not the sense data. Momentum is a concept that we devise that explains the sense data. The data is not predictive, but the abstraction that we use to explain the data is predictive.

The process of going from the data, to an abstraction, to a prediction, to an experiment, to more data is essential to science. And that abstraction is an explanation. What you call a description is, in fact, more than a description. It goes beyond merely reporting what was observed, and provides an explanation for what was observed which can be used to predict what will be observed.

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• Hornbein, nasu and vanhees71
"Abstraction" is the way humans experience the implementation of "economy" (unification). What is abstraction other than a perception, an idea? Economy is the subsumption of a larger set to a smaller set of data (not necessarily sense data) through the rules of logic (which are also data but not sense data). So I would think that economy, having a concrete definition, is a better word than abstraction, hence "economical descriptions" in my proposed definition of scientific knowledge above.

You correctly point out that "momentum" is not sense data, however, I never said that descriptions are sense data! Descriptions rely on internal perceptions. You state that a "description is more than a description" and to support this you repeat what I said about predictions and applications which is how humans utilize those descriptions (Quote: Science provides logically consistent, economical descriptions of our sense data that lead to predictions and applications). Predictions and applications just show how those descriptions are useful and why people like them and have been pursuing them in the first place. They make up for the lack of a crystal ball ...

Regarding symmetry and action-reaction: It is indeed true that translational symmetry is more readily believable than action-reaction. But the former does not imply the latter. We must make stronger assumptions about the nature of forces between two isolated particles to arrive at action-reaction. In particular, we need to assume that the forces are conservative in a quite specific sense. Noether's theorem has these additional assumptions baked into it, so, it is cannot be used to derive action-reaction in the form of conservation of momentum or any other form. That would be cheating :) since it would be like trying to derive something when a big part of it is already assumed. And if one wants to arrive to action-reaction starting from assumptions regarding the symmetry and the conservative nature of forces, this can be done with no use whatsoever of Noether's theorem or of the concept of momentum.

Action-reaction is no less believable than the assumption that internal forces are given by the gradient of a single function of rather special form. But the reduction of one problem to another is very interesting, no question. That's what we do. The pattern/relations is the knowledge, and utility the only validation of the enterprise.

My comments mean no disrespect; you input is apparently based on a profound knowledge of the issues.

• weirdoguy and Dale
You state that a "description is more than a description"
Please do not misquote me. If you are going to quote me use the forum's quote feature so that you can identify what I actually said and others can see it in context, which is:
What you call a description is, in fact, more than a description.

Predictions and applications just show how those descriptions are useful and why people like them and have been pursuing them in the first place.
No matter how you slice it, a prediction requires more than a description. A description only tells you what actually happened. A prediction is a statement (hypothesis) about what would happen in some hypothetical scenario that has not actually happened. There is simply no way to get a prediction from nothing more than a description. You are misusing the word "description", as I pointed out with my above quote.

You will find communication much more effective if you use words correctly. In order to support your claim that science does not explain anything you are misusing the word description. You are taking what a scientific explanation does (make predictions) and trying to hide it in "description".

We must make stronger assumptions about the nature of forces between two isolated particles to arrive at action-reaction. In particular, we need to assume that the forces are conservative in a quite specific sense. Noether's theorem has these additional assumptions baked into it,
I agree. Noether's theorem requires that the laws of physics can be expressed as a Lagrangian. That is the "specific sense" that you mention. The Lagrangian formalism does most of the heavy lifting in Noether's theorem.

it is cannot be used to derive action-reaction in the form of conservation of momentum or any other form. That would be cheating :) since it would be like trying to derive something when a big part of it is already assumed.
This is not correct. It is easy to write down a Lagrangian which does not have conservation of momentum. So we are not assuming the conservation of momentum merely by using the Lagrangian formalism.

Where your point is valid is that when we are talking about explanations based on generally accepted assumptions the assumption that the laws of physics can be written in the Lagrangian formalism is a big one. While it is generally accepted and non-controversial for physicists, it may not be so acceptable for non-physicists. We would do good to recall that, but to call it "cheating" is unjustified. It is valid reasoning for the intended audience of this forum.

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A unified description can be an equation that brings a pattern into focus in place of a long table of data. That may lead to predictions to be tested by experiments. Despite the clarity it offers and its useful ramifications and implications, it is just a description, no matter how you slice it, albeit a useful one. I understand the desire and even the practicality of investing on it further as a scientist but at the end of the day my Occam's razor can't see how description in its common usage of the word is too simple a concept to capture the character of modern scientific knowledge. This is a philosophical problem and I tend to side with those that consider unification (economy) as the only feature of scientific explanation. I am OK with disagreeing with you on that since philosophy is not exactly science.

Let me quote myself because I think you misunderstood me:
"It cannot be used to derive action-reaction in the form of conservation of momentum or any other form. That would be cheating :) since it would be like trying to derive something when a big part of it is already assumed." The "big part" stands for the assumption that forces are given by a potential energy. This is not sufficient for conservation of momentum. However it is necessary, if the plan is to invoke spatial invariance to show action-reaction. I can easily write down translationally invariant Newtonian equations of motion for an isolated pair of particles which manifestly violate action-reaction and thus conservation of momentum even if the forces are given by potential energies! (as long as the potential functions for the two forces are different).

The form of Noether's theorem that can relate spatial symmetry to action-reaction (i.e., conservation of momentum) relies on Lagrangian equations for conservative forces. There is a problem with this:
Spatial symmetry does not imply action-reaction (i.e. conservation of momentum of the particles) through Noether's theorem unless internal forces are derivatives of a common potential function. This is a strong assumption which is not always true in the physical world.

I hope my reasoning is clear to physicists unless non-physicists are more willing to question assumptions.

A clarification: One can argue that all microscopic forces are conservative however, at that scale, one needs to include electromagnetic fields along with matter even in a non-quantum description and so action-reaction is generally violated (although momentum is still conserved if fields are included). There might be an argument starting from conservative microscopic forces leading to action-reaction for non-conservative macroscopic forces. Until this derivation is made explicit, I must take action-reaction to be a fundamental law of Newtonian mechanics that cannot be derived by translational symmetry. In fact, it is the only law (along with one more, subsidiary one) since the first and the second law require that "force" is a primitive concept, which has been debunked 200+ years ago, and therefore they are not laws at all ...

This is not correct. It is easy to write down a Lagrangian which does not have conservation of momentum. So we are not assuming the conservation of momentum merely by using the Lagrangian formalism.
That's of course true, but one must say that these Lagrangians always describe not a fundamental system but some "effective theory". E.g., you can approximately treat the motion of the planets around the Sun of our solar system as the motion of the planet in the fixed gravitational field of the Sun. Then of course momentum conservation is not fulfilled, because translation invariance is broken, because the position of the Sun is now fixed and distinguished from all other points. You still have time-translation invariance (i.e., the Lagrangian doesn't explicitly depend on time) and rotational invariance around the position of the Sun (when treating the gravitational field as the central one of a "point mass", which is of course another approximation since the Sun is not exactly spherically symmetric).

On the other hand, the Lagrangian or Hamiltonian formalism of the least-action principle makes it easy to determine, how the dynamical laws look like given symmetries, particularly the spacetime symmetries of Newtonian mechanics, i.e., invariance of the physical laws under the connected part of the Galilei group, which is a 10-parameter Lie group and thus implies 10 conserved quantities a la Noether: energy (1), momentum (3), angular momentum (3), and center-of-mass velocity (3).

Indeed, when treating the Sun-planet problem as a closed two-body problem of "point masses", all 10 conservation Laws are fulfilled, because it obeys the full Galilei symmetry of Newtonian spacetime.
Where your point is valid is that when we are talking about explanations based on generally accepted assumptions the assumption that the laws of physics can be written in the Lagrangian formalism is a big one. While it is generally accepted and non-controversial for physicists, it may not be so acceptable for non-physicists. We would do good to recall that, but to call it "cheating" is unjustified. It is valid reasoning for the intended audience of this forum.
Well, physics is an empirical science, and so far the formulation of the dynamical laws in terms of the action principle is, on the fundamental level, very successful. In fact all of physics is formulated in terms of the action principle for various fields. For the known elementary particles and all interactions except gravity, it's even a fully quantized quantum field theory. Gravity is described by a classical (i.e., non-quantum) field theory, which also is most conveniently derived from the Hilbert action. In fact Hilbert got the correct Einstein-field equations even a bit earlier in 1915 than Einstein himself, using the action principle ;-)).

"Abstraction" is the way humans experience the implementation of "economy" (unification).

While what will follow is off the topic it is nevertheless interesting. Basketball star Larry Bird and golf star Sam Snead didn't use abstraction. Larry remembered every play in basketball. Larry could watch a play from a game film from years prior and tell you what game it was and which play preceded and followed it. Sam could do the same with every golf shot he had ever taken. This seems to have been an advantage to them. Larry could remember a similar play from the past and so pass to a teammate even though unable to see him. Sam could remember whatever might have been imperfect about some similar shot from the past and correct for it in the present.