# Problem understandinging laws of motion

• Kolahal Bhattacharya
In summary, the most fundamental law of motion according to the conversation is conservation of linear momentum, which is a consequence of symmetry alone. This law holds good independent of the concept of inertial frames and F=ma. While Newton's three laws are all equally fundamental, they do not contain much informational content and it may be more useful to axiomatize physics starting with symmetry rather than Newton's laws. Additionally, Newton's first law provides a framework to define force and mass in a reasonable way.
Kolahal Bhattacharya
I am thinking the following question for a considerable period. Which law of motion of Newton is the most fundamental? ...Well, there are talks and suggestions that it is a non-sense question. However, it may be…I have thought over it and expecting expert’s view. The current situation is that I have collected the following ideas-
Newton’s 3rd law is valid in any frame, inertial and non inertial, it is independent of the concept of inertia. Right now, I am not giving the supporting examples, but they are there.
According to one of my college professors, ‘1st law cannot be derived from 2nd law, because, 2nd law assumes the concept of inertial frame, and hence, that of inertia; i.e. in effect, it assumes the 1st law’. So, it is tempting to say 1st law which serves the concept of inertia, is independent of the 2nd and hence, more fundamental…well, there is a problem. When we say one frame to be inertial, we mean to say every force involved there has physical cause and effect and where F=ma is applicable. So, it is seen that though 1st law gives the concept of inertia, it is 2nd law is also required to define an inertial frame. So, my professor’s argument seems to be invalid. So, what is the correct concept? Also, am I right in thinking that since 3rd law is independent of 1st law or, 2nd law, it is more fundamental? For, in effect, it seems that this law of physics is independent of the frame.
Now, let me justify that truly, 3rd law is independent of 1st and 2nd laws.
Let me start with 3rd law and 1st law. My plan to check whether 3rd law is independent of 1st law and 2nd law is to see
i) When 3rd law is valid, in which frame we are.
ii) When it is not valid, in which frame we are.

i) When 3rd law is valid we are in an inertial frame; but we may also be in a non-inertial frame! Since 3rd law deals with only interaction forces (as per its statement) [still we do not know the interaction is real, 'interaction force' only means one can find who applies the force on whom and how is himself affected] even in an accelerating frame, these forces satisfy 3rd law. After we know about frames and forces, we can identify them to be real forces.

ii) When it is not valid, we cannot conclusively say that we are in a non-inertial frame. There are cases in electrodynamics where in an inertial frame non-relativistic charge particles do not satisfy 3rd law in spite of their real interaction. You may choose to read Griffiths, 8.2: Momentum.

Note, we need not use 2nd law for calculating force magnitudes, 3rd law gives them ready- made to us.

These considerations suggest that 3rd law holds good independent of the concept of inertial frames and F=ma.

It is difficult to show 1st law to be dependent on 3rd law because, the motivation behind 1st law came from the law of inertia due to Galileo. However, conservation of liner momentum follows directly from the 3rd law and vice versa. this conservation principle is most sacred.(For example, in the case of electrodynamics cited above, it is linear momentum which being conserved rescues us from danger).It is interesting that it acts on systems being independent of frames. A possible conformation is that 1st law intuitively follows from the conservation of linear momentum. It is truly very exciting view: the content of 1st law and the conservation principle is the same.

The most fundamental law of motion is conservation of momentum which is a consequence of symmetry alone.

You are absolutely right.But it is not the answer of my question.

actionintegral said:
The most fundamental law of motion is conservation of momentum which is a consequence of symmetry alone.

So is every conservation law. This is not a good statement.

Newton's Three Laws are independent, and so they're all equally fundamental. You need 1 to define 2 and 3, but without 2 you cannot talk about 1 or 3, etc.

Kolahal Bhattacharya said:
You are absolutely right.But it is not the answer of my question.

In a way, it is. Newton's laws don't contain much in the way informational content. For example F=ma is devoid of content. "Force" and "mass" are both undefined terms. So how is it useful to equate them? Acceleration is the only independent quantity in that statement. So if you want to axiomatize physics, you need to start with symmetry , not Newtons's laws.

actionintegral said:
In a way, it is. Newton's laws don't contain much in the way informational content. For example F=ma is devoid of content. "Force" and "mass" are both undefined terms. So how is it useful to equate them? Acceleration is the only independent quantity in that statement. So if you want to axiomatize physics, you need to start with symmetry , not Newtons's laws.

I would argue with that statement, in that Newton's laws provide a perfectly good framework to define "force" and "mass" in a perfectly reasonable way.

Newton's first law says that an object in the absence of forces will sit in an inertial reference frame. Inertial reference frames are defined as frames not under acceleration, and you can measure acceleration experimentally (if you believe Newton's first law). So you take some object and put it somewhere and holy crap it accelerates! I can measure this acceleration, and write it down. Now let's say that we know that this motion is due to electromagnetic interactions...so then we put down another mass with the same charge and see what happens. We can calculate the relative masses of any number of objects doing this, giving a perfectly good definition of mass.

As for the assertion that force is ill-defined, this is untrue. It's just what is exerted on an object to cause its self-frame to become non-inertial. Newton's laws are theoretically complete, it's just that it's usually easier to analyze dynamics in the context of other formulations, that make exploiting symmetries easier.

StatMechGuy said:
I can measure this acceleration, and write it down.

But that is what I am trying to say. Can you measure the force? No. Can you measure the mass? No. You can only measure the acceleration.

Now two identical objects stuck together will cause more momentum change than a single object. I haven't been able to demonstrate that that follows from symmetry alone, but I'm not the brightest bulb.

actionintegral said:
But that is what I am trying to say. Can you measure the force? No. Can you measure the mass? No. You can only measure the acceleration.

Now two identical objects stuck together will cause more momentum change than a single object. I haven't been able to demonstrate that that follows from symmetry alone, but I'm not the brightest bulb.

No, you can measure the acceleration on two different objects subjected to the same force, and measure the ratio of the accelerations, and that gives you a standard of mass. Then you can go to other forces with your relative mass in hand and work out those forces from a standard force. You just define a standard of force that you know to be the same for two objects and run from there.

The conservation laws being a consequence of symmetry is a result of Noether's theorem, which in turn is a result of Lagrangian formalism, which is derived from Newton's laws.

StatMechGuy said:
The conservation laws being a consequence of symmetry is a result of Noether's theorem, which in turn is a result of Lagrangian formalism, which is derived from Newton's laws.

Or vice versa. And now I think we have reached a point of common agreement.

Re, Newton's law

Let me carry with the things I found uncomfortable in your posts.

StatMechGuy1.If those are equally fundamental,and independent of each other, how can you say that 1 is needed to have 2 and 3?etc.

Actionintegral2.I agree with you that F is undefined in Newton's laws.I also say if I try to axiomatize physics,I need to start with symmetry concepts... But these do not anyway help to say which law of motion of Newton is more fundamental.

StatMechGuy2.a]Newton's law of inertia gives the definition of inertial mass, though it is not related to our topic, it is not a perfect definition of mass.b]What you say as the definition of force, mentioning frames, depends on the concept of frames and hence, on force themselves!Newton's laws do not define force.We already have what force is.It develops from interaction and affects the applied body,or particle, in changing its state of motion(we still do not know of frames) and may be some other aspects...Newton's laws make our intuitive concept of force more concrete.

Finally, I would say though I must appreciate the later part of your posts,it confirms nothing regarding my question.Well, one may tempted to say these laws are equally fundamental.But,I have a proposal here.Think of force as it is developed from interaction.In Newtonian domain, we have nothing to counter this and the law of this interaction is provided by 3rd law.Once we know how force is developed, we may be interested in finding
its effect.When it is applied (as one of the pair-force)on classical objects, their response,depending on their property called inertia(1st law),comes out(2nd law).Pseudo force is not a force as Newtons laws believe.So, there cannot be any question regarding these things in my problem.So, if there is not any most fundamental one, a systematic development may be thought from the 3rd law towards the laws of inertia.Personally, I cannot separate 1st and 2nd law regarding concept of inertia.But, 3rd law is independent of the concept of inertia.

Kolahal Bhattacharya said:
interaction is provided by 3rd law.

KB, I guess I would agree with you here. The third law says that there is an undefined thing called an "action". We know nothing about it except that is always is matched by a balancing "reaction". So that the total of these two is always zero.

But my disagreement with you lies elsewhere. I don't think these laws were intended to be read like Euclid's postulates. I think they were intended as heuristic arguments spoken in the language of the day to counter the prevailing thoughts of the day (especially the first law).

If you want to build the universe with postulates - start here:

1. There is a thing called a "light corpuscle" which is travels at the same for all observers. (Einstein)

2. There is a thing called "mass" which pulses with frequency mc*c = h*nu. (debroglie)

And off you go...

Kolahal Bhattacharya said:
Let me carry with the things I found uncomfortable in your posts.

StatMechGuy1.If those are equally fundamental,and independent of each other, how can you say that 1 is needed to have 2 and 3?etc.

Actionintegral2.I agree with you that F is undefined in Newton's laws.I also say if I try to axiomatize physics,I need to start with symmetry concepts... But these do not anyway help to say which law of motion of Newton is more fundamental.

StatMechGuy2.a]Newton's law of inertia gives the definition of inertial mass, though it is not related to our topic, it is not a perfect definition of mass.b]What you say as the definition of force, mentioning frames, depends on the concept of frames and hence, on force themselves!Newton's laws do not define force.We already have what force is.It develops from interaction and affects the applied body,or particle, in changing its state of motion(we still do not know of frames) and may be some other aspects...Newton's laws make our intuitive concept of force more concrete.

To answer your first point, the three are fundamental but not independent. You have to have all three simultaneously written down or else it's just jibberish. This is not unheard-of. Axiomatic set theory is dependant on several postulates that must simultaneously be written down to have a theoretically "complete" theory.

Also, Newton's laws define inertial mass, but that's the only kind of mass there is. Even in relativity, the gravitational mass is the same as the inertial mass, and it provides a way of axiomatizing the concept of mass.

I most certainly can talk about reference frames without talking about force. I can put down a grid of rods and say "this is a reference frame". Then I can take those rods and move them at constant velocity and say "this is a reference frame". I can measure acceleration by observing, as per Newton's first law, that if I put two objects together and put the grid of rods around each one and compare those grids, that one is accelerating relative to the other. If I find that one of the object's grids is accelerating relative to every other reference frame, then clearly this is not an inertial reference frame. The second law tells you how the non-inertial reference frame of an object evolves under some external force (defined through the first law), and the second law provides a way of testing if I'm in an inertial reference frame (I see forces where I know there aren't any). The third law is kind of odd, but it gives us a way of comparing how two non-inertial reference frames evolve relative to each other.

Re, Newton's law

1.action integral:Well, building a universe might be too heavy a task for me!
2.statmechguy:Well, you can certainly say reference frames in a kinematic way so that your definition of force does not depend on itself.These frames may give you the velocity, acceleration etc;but, do not anyway help you to define forces.To have the frame which gives the concept of force, you need to have inertial frames.Inertial frames, in the sense of dynamics, is defined by both 1st and 2nd law.To quote A.P.French, 'The "best" choice of reference frame is ultimately a question of dynamics'.And you call that frame inertial where you would see 2nd law to be valid.Lastly, your saying regarding 3rd law and frames is not understood.3rd law IS independent of the concept if inertial frame.How does the evolution of frame occurs in 3rd law?

StatMechGuy said:
I most certainly can talk about reference frames without talking about force. I can put down a grid of rods and say "this is a reference frame". Then I can take those rods and move them at constant velocity and say "this is a reference frame". I can measure acceleration by observing, as per Newton's first law, that if I put two objects together and put the grid of rods around each one and compare those grids, that one is accelerating relative to the other.

Up to here, I agree. However...

If I find that one of the object's grids is accelerating relative to every other reference frame, then clearly this is not an inertial reference frame.

I can define frames which are in uniform motion wrt a non-inertial frame. In fact, there are just as "many" grids in uniform motion wrt a non-inertial frame, as there are grids in uniform motion wrt an inertial frame (and are hence inertial frames themselves).

The difficulty with "picking out" what is an inertial frame is related to saying when "an object is not subject to a force", and then later you are going to use this frame to find out if others are, or aren't, subject to a force. So there is some kind of circularity here.

To illustrate this, consider a toy universe with 3 matter points in it. Now, define a frame to each of the matter points (and let's leave out the other difficulty of rotation - assume that you can "point to the distant stars" or something). Let's say that we observe that each of the two matter points is accelerating in the third point's frame. (imagine them for instance subject to an overall uniform electric field, them being slightly charged, and also having interactions amongst themselves)
It is now entirely arbitrary to "pick" one frame, say that it is an inertial frame, and say that the particles accelerating in it are subject to a "force".
One will find different forces as a function of this random choice of "inertial frame'.

Re, Newton's law

How is your post related to my problem?Forgive me, I did not see the relevance.

Kolahal Bhattacharya said:
1.action integral:Well, building a universe might be too heavy a task for me!
2.statmechguy:Well, you can certainly say reference frames in a kinematic way so that your definition of force does not depend on itself.These frames may give you the velocity, acceleration etc;but, do not anyway help you to define forces.To have the frame which gives the concept of force, you need to have inertial frames.Inertial frames, in the sense of dynamics, is defined by both 1st and 2nd law.To quote A.P.French, 'The "best" choice of reference frame is ultimately a question of dynamics'.And you call that frame inertial where you would see 2nd law to be valid.Lastly, your saying regarding 3rd law and frames is not understood.3rd law IS independent of the concept if inertial frame.How does the evolution of frame occurs in 3rd law?

This thread is really an interesting one with some fundamental discussion going on Newton's laws and well I also believe that Newton's 3rd Law has something deeper to say as it is the only law which doesn;t find much generalization in physics as the other two laws were carried over from Newtonian Physics to Relativity physics, also I would like to bring attention the Relativity Principle:
Forms of laws of physics( and not just Laws of motion) will remain invariant in inertial frame of references.

So inertial frame of references are not just the one where 2nd law holds, for other laws of physics without force or motion into consideration can also define inertial frame if we see that in a particular frame of reference the laws are invariant under Lorentz transformations for all laws we can consider besides the 2nd Law, so that is the beauty of the Relativity Principle underlying Special Relativity of it being independent of any physical law but more general statement of laws of physics.

Re, Newton's law

Well, I see and acknowledge.

## 1. What are the three laws of motion?

The three laws of motion, also known as Newton's laws of motion, are a set of principles that describe how objects move in relation to the forces acting upon them. The first law states that an object at rest will stay at rest and an object in motion will stay in motion unless acted upon by an external force. The second law states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. The third law states that for every action, there is an equal and opposite reaction.

## 2. Why are the laws of motion important?

The laws of motion are important because they form the basis of classical mechanics and are fundamental principles that govern the behavior of objects in motion. They are used to explain and predict the motion of objects in everyday life and in scientific fields such as physics, engineering, and astronomy.

## 3. How were the laws of motion discovered?

The laws of motion were discovered by Sir Isaac Newton in the late 17th century. Newton formulated these laws based on his observations and experiments on the motion of objects. He published them in his famous work "Principia Mathematica" in 1687.

## 4. Can the laws of motion be broken?

No, the laws of motion cannot be broken. They are fundamental principles of physics that have been tested and verified through numerous experiments and observations. However, there are certain situations, such as extreme speeds or in the presence of strong gravitational forces, where the laws of motion may appear to be violated, but they still hold true.

## 5. How do the laws of motion relate to other scientific principles?

The laws of motion relate to other scientific principles such as energy conservation, momentum conservation, and the law of universal gravitation. These principles work together to explain the behavior of objects in motion and how they interact with each other in the physical world.

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