# Can Newton's Laws of Motion be simplified?

## Main Question or Discussion Point

I was just thinking, are Newton's laws of motion most fundamental, or can they be reduced to something even simpler?

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phinds
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I was just thinking, are Newton's laws of motion most fundamental, or can they be reduced to something even simpler?
Ya reckon maybe someone would have already done it if it could be done?

Andrew Mason
Homework Helper
I was just thinking, are Newton's laws of motion most fundamental, or can they be reduced to something even simpler?
I think they can be derived from more general principles. Whether that makes them simpler or not, depends on your point of view.

It can be shown that Newton's second and third laws follow from the concept of Galilean relativity: 1. that all inertial frames of reference are equivalent in the sense that the laws of physics are the same to all inertial observers and 2. that time and distance are measured to be the same by all observers.

With these principles, you can derive the second and third laws of Newton. Or you can start with Newton's laws (and the assumption that time and distance are the same to all observers) to derive the principles of Galilean relativity.

AM

I think they can be derived from more general principles. Whether that makes them simpler or not, depends on your point of view.

It can be shown that Newton's second and third laws follow from the concept of Galilean relativity: 1. that all inertial frames of reference are equivalent in the sense that the laws of physics are the same to all inertial observers and 2. that time and distance are measured to be the same by all observers.

With these principles, you can derive the second and third laws of Newton. Or you can start with Newton's laws (and the assumption that time and distance are the same to all observers) to derive the principles of Galilean relativity.

AM
Thanks for that answer. Reading that gave me a kind of 'eureka!' moment.

But where does "equal and opposite" come from in Galilean relativity?

Does the 3rd Law still hold true in Special Relativity?

But where does "equal and opposite" come from in Galilean relativity?

Does the 3rd Law still hold true in Special Relativity?
no,it does not but if you are careful you can count it.just think about two charges whose direction of motion are perpendicular to each other and see if you can verify newton third law.however if you will be careful you will be but in a direct way you can not.

I think they can be derived from more general principles. Whether that makes them simpler or not, depends on your point of view.

It can be shown that Newton's second and third laws follow from the concept of Galilean relativity: 1. that all inertial frames of reference are equivalent in the sense that the laws of physics are the same to all inertial observers and 2. that time and distance are measured to be the same by all observers.

With these principles, you can derive the second and third laws of Newton. Or you can start with Newton's laws (and the assumption that time and distance are the same to all observers) to derive the principles of Galilean relativity.

AM
Wow, I would like to see those derivations - I never knew that was possible. What should I search for?

Andrew Mason
Homework Helper
But where does "equal and opposite" come from in Galilean relativity?
Galilean relativity follows from the third law. So if you go backward and assume Galilean relativity as a premise, the third law follows.

Start with the first law and the equivalence of all inertial frames and the equivalence of time measurements. If I am in an inertial frame defined by a body A on which no forces act, there can be no change in its state of motion relative to any other inertial frame. This means there can be no change in momentum of A.

Let's then suppose that A is a black box that is full of little balls bouncing around inside the box. Since there is no change in A's momentum, then the sum of all forces between the little balls multiplied by the time over which they act (ie. the impulses or change of momentum) must equal zero.

The only way this can occur is if the forces in any interaction between two balls are equal and opposite OR there is a difference in the measurements of the time over which an interaction between two balls occurs that are made by each of the two interacting balls. Since our premise is that measurements of time and distance are the same in all frames of reference the latter cannot be the case. So, the only way that momentum can be conserved is if the forces are equal and opposite in any interaction.

Does the 3rd Law still hold true in Special Relativity?
The conservation of momentum does hold. But the concept of simultanaeity disappears. Absolute simultanaeity is implicit in the third law. Since the times of interaction are not the same between interacting bodies moving at relativistic speeds, the forces are not equal and opposite same so the third law does not hold in SR.

AM

Andrew Mason
Homework Helper
Wow, I would like to see those derivations - I never knew that was possible. What should I search for?
You could google "galilean relativity". You will find discussion of Galilean relativity following from Newton's laws but you will have a difficult time finding anyone who discusses the reverse connection.

Newton's laws follow from Galilean relativity just as Galilean relativity follows from Newton's laws. They are equivalent ways of describing the laws of physics (where time and distance measurements are the same in all frames of reference). If Newton's laws do not hold, Galilean relativity does not hold. If Galilean relativity is incorrect, Newton's laws are incorrect.

In my previous post I outlined how the third law follows from Galilean Relativity. The second law is a little more subtle:

Start with the premise that all inertial frames are equivalent. This means that the same phenomenon occurring in one inertial frame must have the same effect if it occurs in any other inertial frame. Otherwise, the frames would not be equivalent.

So, for example, if I apply a unit of "pull" (say a standard spring stretched to a unit distance) to a unit of matter for a unit of time in one inertial frame it will have the same effect in any other inertial frame. This necessarily means that acceleration must be constant because if I do two such pulls sequentially the second pull must have the same effect as the first (i.e. in the inertial frame of reference of the body after the first pull ends).

Also, if I perform two unit pulls in tandem on each of two unit bodies for a unit of time simultaneously, I should get the same change in motion as when I just apply one unit pull to one unit body for a unit of time and then apply another unit pull to the other body. They will end up in the same final inertial reference frame whether I do it simultaneously or sequentially. Therein lies the concept of mass (think of a unit of mass as a neutron or electron-proton pair and a body as an aggregation of such units and its mass the number of such units contained in the body).

AM

The conservation of momentum does hold. But the concept of simultanaeity disappears. Absolute simultanaeity is implicit in the third law. Since the times of interaction are not the same between interacting bodies moving at relativistic speeds, the forces are not equal and opposite same so the third law does not hold in SR.

AM
I'm not sure I follow this.

If we are talking about the kinds of interactions that basic physics classes talk about (balls and boxes and the like), wouldn't any interaction require the distance between the interacting objects be zero, and therefore wouldn't the times of interaction be equal regardless of reference frame?

Andrew Mason
Homework Helper
I'm not sure I follow this.

If we are talking about the kinds of interactions that basic physics classes talk about (balls and boxes and the like), wouldn't any interaction require the distance between the interacting objects be zero, and therefore wouldn't the times of interaction be equal regardless of reference frame?
If you are talking about balls and boxes, you would be right - although the distance between interacting surfaces is not zero - just very small. Any "collision" involves atoms interacting through electrical forces at a distance.

If you are talking about collisions of protons travelling at .999999c relative to each other and colliding with each other, the times of interactions for both proton would not be the same as measured in the frame of reference of one of the protons.

AM

K^2
Does the 3rd Law still hold true in Special Relativity?
It does if you are careful about how you define forces and acceleration. A better statement would be that momentum is still conserved in Special Relativity. So if you define forces via dp/dt, 3rd law must hold.

D H
Staff Emeritus
It can be shown that Newton's second and third laws follow from the concept of Galilean relativity: 1. that all inertial frames of reference are equivalent in the sense that the laws of physics are the same to all inertial observers and 2. that time and distance are measured to be the same by all observers.
I disagree. Galilean relativity is a consequence of Newton's first law (inertial frames exist) and Newton's second law (F=ma; force and acceleration are frame invariant). Not the third law. That is a beast of a quite different color.

Newton's third law is a true law of physics. It's empirical, it's not always true, and it can be derived from deeper concepts. Those deeper concepts are conservation of linear and angular momentum. Newton's third law assumes that individual forces can always be paired and that force transmission is instantaneous (action at a distance). With multi-body forces such as the chiral three-nucleon forces in a helium nucleus you don't get Newton's third law. If the force transmission isn't instantaneous, the field that mediates the force will itself contain linear and angular momentum, and Newton's third law once again fails to be true.

With those simplifying assumptions, conservation of linear momentum yields the weak form of Newton's third law, that forces exerted by a pair of interacting particles on one another are equal in magnitude but opposite in direction. Adding conservation of angular momentum yields the strong form of Newton's third law, that forces exerted by a pair of interacting particles on one another are equal in magnitude but opposite in direction, and are directed along the line connecting the two particles.

There's something even deeper than the conservation laws, and that's Noether's theorems. However, the conservation laws (and even deeper, Noether's theorems) aren't simplifications of Newtonian mechanics. They're more complex. Newton's laws are the simple form.

Andrew Mason
Homework Helper
It does if you are careful about how you define forces and acceleration. A better statement would be that momentum is still conserved in Special Relativity. So if you define forces via dp/dt, 3rd law must hold.
But you would have to measure momentum, force and time all in the same frame of reference. Newton's third law would apply if you did that. But Newton's third law talks about the force of A on B and the force of B on A being equal. If these forces are measured in the respective reference frames of each body, they will necessarily differ because time measurements will differ between frames.

AM

K^2
Yes, if you measure the two in different frames of reference, you'll end up with a discrepancy. That's why I said you have to be careful about how you define forces.

Andrew Mason
Homework Helper
I disagree. Galilean relativity is a consequence of Newton's first law (inertial frames exist) and Newton's second law (F=ma; force and acceleration are frame invariant). Not the third law. That is a beast of a quite different color.
That is one way of looking at it. One could also say that Newton's first and second law PLUS the premise that time and distance measurements are equal will lead to the third law. Then one could say that the second and third laws (with that premise) imply Galilean relativity.

The third law is a necessary consequence of Galilean relativity (the equivalence of all inertial frames if time measurements are equal in all frames). Consider the frame of reference of the centre of mass of two bodies moving toward each other on a collision course. Since our premise is that time measurements are the same for each body, the duration of the collision is measured to be the same in all frames. So, if the collision between the two bodies did not produce equal and opposite forces on the respective bodies at all times, total momentum would not be conserved. Consequently the centre of mass of the two body system would change its motion (ie the inertial frame of reference of the centre of mass before and after the collision would be different) and that would violate Galilean relativity - an inertial frame of reference changing its motion without any external force affecting it.

Newton's third law is a true law of physics. It's empirical, it's not always true, and it can be derived from deeper concepts. Those deeper concepts are conservation of linear and angular momentum. Newton's third law assumes that individual forces can always be paired and that force transmission is instantaneous (action at a distance). With multi-body forces such as the chiral three-nucleon forces in a helium nucleus you don't get Newton's third law. If the force transmission isn't instantaneous, the field that mediates the force will itself contain linear and angular momentum, and Newton's third law once again fails to be true.
I agree that the conservation of momentum is fundamental. However, it is not just Newton's third law that fails to hold if force transmission is not instantaneous. The second also does not hold. The second law implies that masses of bodies do not change in an interaction. However, if forces at a distance do not act instantaneously, the conservation of momentum requires, as SR provides, that energy carries momentum. As a result, when a collision is measured in a single inertial reference frame (e.g. the centre of mass frame), mass is transferred between bodies due to the transfer of energy.

With those simplifying assumptions, conservation of linear momentum yields the weak form of Newton's third law, that forces exerted by a pair of interacting particles on one another are equal in magnitude but opposite in direction. Adding conservation of angular momentum yields the strong form of Newton's third law, that forces exerted by a pair of interacting particles on one another are equal in magnitude but opposite in direction, and are directed along the line connecting the two particles.

There's something even deeper than the conservation laws, and that's Noether's theorems. However, the conservation laws (and even deeper, Noether's theorems) aren't simplifications of Newtonian mechanics. They're more complex. Newton's laws are the simple form.
All I am saying is that Galilean relativity is an even simpler form. Newton's laws flow from the premises of Galilean relativity: if it were not the case that F=ma or if interaction forces were not equal and opposite, (assuming time and distance measurements to be the same in all frames of reference) Galilean relativity would be violated.

AM

It is possible to derive Newton's laws from Hamilton's principle. Hamilton's principle states that the time evolution of a system that obtains physically is the one which minimizes (or maximizes, in certain cases) the time integral of the Lagrangian, which is equal to the difference between kinetic and potential energies.

Also, there seems to be a lot of argument in this thread about whether Newton's first and second laws are actually "physics" or whether they are merely mathematical definitions (respectively for inertial frames and force). There was a thread a few weeks ago which went pretty in-depth on this argument. https://www.physicsforums.com/showthread.php?t=625248

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Newton's first law that an object not acted upon by a force maintains a uniform velocity is really a special case of his second law F = ma, in the special case that F= 0.

So there really are only two independent Newton's Laws.

Andrew Mason: I don't like your derivation of the third law from Galilean relativity. Suppose that we have a system of three particles A,B,C. Then the condition that the momentum is conserved in the system is as you say:
(Fab+Fac+Fba+Fbc+Fca+Fcb)*dt=0
But I don't see how setting Fxy = -Fyx is the only solution to that equation. How about for instance: Fac = -Fab, Fba=-Fbc,Fca=-Fcb? why couldn't I assume that. It would be a wicked world yes, but I don't see a problem with it.

Andrew Mason
Homework Helper
Andrew Mason: I don't like your derivation of the third law from Galilean relativity. Suppose that we have a system of three particles A,B,C. Then the condition that the momentum is conserved in the system is as you say:
(Fab+Fac+Fba+Fbc+Fca+Fcb)*dt=0
But I don't see how setting Fxy = -Fyx is the only solution to that equation. How about for instance: Fac = -Fab, Fba=-Fbc,Fca=-Fcb? why couldn't I assume that. It would be a wicked world yes, but I don't see a problem with it.
The interaction times are not the same for all the interactions.

If, during an interval Δt, a, b and c interact with each other, the times of interactions between a and b, between a and c and between b and c will almost always differ. You have to analyse each interaction separately. When you do that, the following will always hold true in Galilean Relativity:

(Fab + Fba)Δtba + (Fac + Fca)Δtac + (Fbc + Fcb)Δtbc = 0

In order for this relationship to apply at ALL times during the interaction, the force pairs must be equal in magnitude and opposite in direction (ie. each force pair sums to 0).

AM