# The need of Newtons first law of dynamics

## Main Question or Discussion Point

Newtons first law states that an object will continue to remain in its state of rest or motion with uniform velocity unless and until acted upon by a net external force.
Newtons second law F=ma
In this equation if F=0 then a=0 that is the same as the first law...
So, if the first law is a special case of the second why is the first law needed ?

Thanks

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Newton´s first law is not really needed, just as you say. I think historically Galileo Galilei had formulated the first law, "the law of inertia", before Newton formulated the second law. But as you say, it is unnecessary.

Fredrik
Staff Emeritus
Gold Member
This is how Wikipedia states the first law.
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 persists in its state of being at rest or of moving uniformly straight forward, except insofar as it is compelled to change its state by force impressed.
The English version is supposedly a 1999 translation of Newton's original Latin. Note that this "law" isn't stated in mathematical terms. To incorporate it into a theory of physics, we must translate it to a mathematical statement. But it's not like there's exactly one mathematical statement that obviously corresponds to what Newton wrote. It can be interpreted as saying exactly that F=0 implies a=0. In that case, it's definitely a special case of the second, assuming that the second is interpreted as F=ma.

However, the use of the word "straight" suggests another interpretation. It suggests the existence of a coordinate system (presumably on ##\mathbb R^3## ("space")) in which motion unaffected by forces is represented by a straight line. So a generous interpretation would say that Newton's first law is a statement about the existence of such a coordinate system. That would make it something more than a special case of the second.

A very generous interpretation would say that the first law tells us (or at least suggests) that we should use what is now called "Galilean spacetime" as the mathematical representation of space and time.

Fredrik
Staff Emeritus
Gold Member
By the way, this is how I think of Newtonian mechanics: The first thing we need is a mathematical representation of space and time, a mathematical structure that we will call "spacetime". This is indeed the starting point of all classical theories of space, time, matter and motion up to and including general relativity. We choose Galilean spacetime because it's consistent with our everyday experiences. This idea corresponds roughly to Newton's first law.

The second thing we need is a way to add matter to an empty spacetime. The "Newtonian" way to add matter to spacetime is to simply postulate the existence of particles whose masses and motion satisfy a differential equation that, given some mild assumptions, has exactly one solution for each initial condition. We choose mx''(t)=F(x(t),x'(t),t) because it meets this requirement. This corresponds to Newton's second law. (The "Lagrangian" and "Hamiltonian" approaches are not different theories, they are slightly different ways to add matter to an empty spacetime).

The third thing we need is some sort of guiding principle that narrows down the list of candidates for the function F. So we postulate that the fundamental forces are always such that the force particle 1 exerts on particle 2 is equal in magnitude and opposite in direction to the force that particle 2 exerts on particle 1. This is of course Newton's third law.

Dale
Mentor
You can also consider Newton's first law in the context of competing theories of physics. There were other theories (Aristotle) that claimed that motion in a straight line at a constant speed required a constant force. He was clearly differentiating his theory from that class of theories.

Thank you

I would actually also say that Newton´s third law is unnecessary. If there are no external forces on a system, the system cannot undergo acceleration. If there is no acceleration of the system than the forces within the system must be balanced by equal but opposite forces...

So you can basically get Newton´s third law from the second as well.

AlephZero
Homework Helper
I would actually also say that Newton´s third law is unnecessary. If there are no external forces on a system, the system cannot undergo acceleration. If there is no acceleration of the system than the forces within the system must be balanced by equal but opposite forces...

So you can basically get Newton´s third law from the second as well.
The third law doesn't say the forces within the system must be balanced. It says there must be equal and opposite pairs of forces.

It's easy to invent a set of more than two forces which are "balanced" but contradict the 3rd law.

Fredrik
Staff Emeritus
Gold Member
The third law doesn't say the forces within the system must be balanced. It says there must be equal and opposite pairs of forces.

It's easy to invent a set of more than two forces which are "balanced" but contradict the 3rd law.
I'd like to see what sort of example you have in mind. All I can think of (at 4:20 AM when my brain feels like...uh...I can't even figure out how to end this sentence) are examples that have particles spinning around in circles or something. Are there any examples that are consistent with the symmetries of Galilean spacetime?

You can also consider Newton's first law in the context of competing theories of physics. There were other theories (Aristotle) that claimed that motion in a straight line at a constant speed required a constant force. He was clearly differentiating his theory from that class of theories.
This, I believe, is the reason it's retained. It functions as the transition statement addressed to the physics naive taking their first steps into physics, not just Aristotelians, but modern everyman [everystudent] who may have some erroneous preconception ["competing theory"] about why things move or don't move. It's the insight that carries them over the threshold from complete lack of rigor in thinking to the realization all this can be rigorously worked out. It's the first statement of rigorous physics most people encounter. It's necessary because they aren't scientists yet.

If it seems like a trivial special case of the second law to anyone they have to consider it took about 2000 years between the time Aristotle first started seriously pondering the cause of motion before Galileo rigorously explained and geometrically proved that nothing sustains motion. Apparently, therefore, it seems virtually hardwired into the human brain to assume a thing in motion is being 'fueled' somehow, and that the fuel eventually runs out. Had Galileo not proven the first law, Aristotelian thinking would have continued to dominate the scene indefinitely.

The first law is there to say what motion isn't. That may sound superfluous, but if the human mind grasped this sort of thing automatically without misinterpretation, we probably wouldn't need physics to begin with. That's my take on why it perpetually stays on the top three list.

AlephZero
Homework Helper
I can't even figure out how to end this sentence) are examples that have particles spinning around in circles or something. Are there any examples that are consistent with the symmetries of Galilean spacetime?
Suppose you have a square plate, surrounded by a circular ring so that the four corners of the square are in contact with the ring, and the ring is a shrink fit around the square.

Suppose the 4 "action" forces of the square on the ring are equal and directed away from the center of the system, but the "reaction" forces of the ring on the square are zero.

Both bodies are in equilibrium and so is the system, but it doesn't satisfy the third law.

Fredrik
Staff Emeritus
Gold Member
I'm not sure I understand this example. Are we to think of these objects as not having component parts? If we think of them as consisting of component parts, then the forces applied to four small parts of the ring would, according to the second law, cause those parts to accelerate unless all the forces on each of those parts add up to zero. So if the ring has component parts, there must be forces between them that pull the parts near the corners of the square towards the center. And if they are being pulled towards the center, then how can the ring not be exerting any forces on component parts of the square?

If we draw the forces on a component of the ring near a corner, we would probably draw three arrows. One for the force that the corner exerts on the ring component, and two for the internal forces in the ring, one on each side. Those arrows could of course be viewed as sums of many smaller contributions, none of which is equal and opposite to the force from the corner on the ring component. But their sum is.

I seem to have drifted into talking about how things actually are, rather than about how they would have to be for Newtons 2nd to hold but not his 3rd. But that's where your example took me, so I guess I just don't see how it can be used as a counterargument to what Agerhell said. To counter what he said, I would use an example like three particles with mass m, forming the corners of an isosceles triangle (not a physical triangle...I'm just describing their relative positions), with the following forces acting on them: At times t such that 0≤t≤1, Particle A is pulling particle B towards particle A...and so on, i.e., B pulls C towards B, and C pulls A towards C. At other times, the force is 0. (I only added that requirement to prevent their speeds from going to infinity, but that's actually irrelevant to the main point). In this scenario, the 3rd is violated, even if the 2nd is not. So this is enough to prove that the 2nd alone doesn't imply the 3rd. However, this example severely violates rotational invariance, or equivalently, conservation of angular momentum, so I'm thinking that Agerhell's argument may work if we add assumptions of symmetry.

D H
Staff Emeritus
You can also consider Newton's first law in the context of competing theories of physics.
This, I believe, is the reason it's retained.
That is one of the key reasons why Newton wrote it. Why it's retained is a different matter. Introductory physics texts go into quite a bit of depth on frames of reference before diving into Newton's laws. Newton's first law is a null test of whether a reference frame is or is not inertial.

Suppose that you see some object accelerating and yet by some means you know that the net force on some object is zero. Does this invalidate Newton's laws? No. It merely means that you are not observing the motion from the perspective of an inertial frame.

To me, Newton's second law is an extension of Newton's first law to regimes where the net force is not null. It is the second law that is subservient to the first, not the other way around. The second law must necessarily be consistent with that already-established first law.

AlephZero
Homework Helper
I'm not sure I understand this example. Are we to think of these objects as not having component parts? If we think of them as consisting of component parts, then the forces applied to four small parts of the ring would, according to the second law, cause those parts to accelerate unless all the forces on each of those parts add up to zero. So if the ring has component parts, there must be forces between them that pull the parts near the corners of the square towards the center. And if they are being pulled towards the center, then how can the ring not be exerting any forces on component parts of the square?
Well, if you postulate one thing that doesn't correspond to "reality" (or "common sense"), it's not surprising that it has consequences that also don't correspond to reality.

If you consider cutting a solid body into parts, the idea that action and reaction are not equal and opposite contradicts the idea of a stress tensor where the force is the dot product of the tensor and the normal direction, because it's hard to imagine how you can have ##\sigma . {\bf n} \ne -(\sigma .({-\bf n}))##.

I was thinking of this in a simpler way. There are two bodies, the ring and the square, and the action and reaction forces between them are at the 4 points where they touch. If the square "expands uniformly" for some reason, by symmetry it will push on the ring with 4 equal radial forces of magnitude P. By symmetry, the ring will push on the square with equal forces of magnitude Q. But if you consider the equilibriom of each body as a whole, there is no reason why P = -Q.

As a "continuous" version of the same situation, you could consider a hollow sphere containing a pressurized fluid. In that case it is clear that the sphere will be in equlibrium for any magnitude of the internal pressure (ignoring the finite strength of the material). It would be a nice engineering trick for designing pressure vessels if you could make the radial force acting on the sphere different from the internal pressure in the fluiid, but the third law says this is impossible.

Fredrik
Staff Emeritus
Gold Member
I was thinking of this in a simpler way. There are two bodies, the ring and the square, and the action and reaction forces between them are at the 4 points where they touch. If the square "expands uniformly" for some reason, by symmetry it will push on the ring with 4 equal radial forces of magnitude P. By symmetry, the ring will push on the square with equal forces of magnitude Q. But if you consider the equilibriom of each body as a whole, there is no reason why P = -Q.
I don't understand. Do you mean that the forces that the ring exerts on the corners on the square could be different in magnitude? That they could e.g. be 10 N at two opposite corners and 20 N at the other two corners? I suppose they could be, but this brings us back to what I said at the end of my previous post: I suspect that Agerhell's argument would work if we add assumptions of symmetry. If you meant something like that the ring doesn't have to push back at all, then I don't understand what a force is in this theory (a theory that respects the 2nd law and violates the 3rd). What would it mean to say that a square exerts forces on a ring at the corners, when the ring by assumption is incapable of responding to that force in any way?

AlephZero
Homework Helper
I wasn't trying to invent a consistent theory that could be applied to any situation - just saying that in a particular situation there can be global equlibrium without local pairs of equal and opposite forces.

But how about something like this (I admit I haven't thought through all the details). Suppose there are two bodies A and B made of materuals with different properties, such that the action force of A on B is concentrated at the point of contact, but the reaction force of B on A is distributed over the whole body of A like some sort of magical force field, and the resultant reacton force of on A is equal and opposite to the action force. That would seem to satisfy the first two laws but not the third.

Dale
Mentor
I would actually also say that Newton´s third law is unnecessary. If there are no external forces on a system, the system cannot undergo acceleration. If there is no acceleration of the system than the forces within the system must be balanced by equal but opposite forces...

So you can basically get Newton´s third law from the second as well.
No. Consider a situation where the interaction forces were equal and opposite, but not colinear. Then Newton's first two laws would be satisfied for all systems but physics would be different (angular momentum would not be conserved).

Ken G
Gold Member
My view is similar to Fredrik's and Dale Spam's, but I'll put it a slightly different way. I see all three laws as essential. The first law is an effort to establish what we would call the inertial path. Newton didn't understand the importance of coordinate-free language, but if we extend to relativity, we can see that the First Law maps into the concept of a geodesic in spacetime-- the inertial path is the path of maximum proper time. If one does not understand the need for proper time, one simply gets Newton's version, but the point is, we still need a way to specify what the inertial paths are first, before we can say what forces do. As mentioned above, the huge revolution in Newton's way of thinking is that dynamics are about what is changing what is happening, not what is happening. (The ancients wanted motion to happen for a reason-- they would be very disappointed to discover that it is not motion, but change in motion, that happens for a reason.)

Then the second law says how forces cause deviations from the inertial path. We cannot say that F=0 means a=0 and we have the first law from the second, because we don't even know what a=0 means without the first law, that's the whole purpose of the first law. Then the third law asserts that the F that causes deviations from the inertial path must be external to the system-- it rules out internally generated forces being the cause of acceleration of a whole system. We cannot say that F=ma already tells us that if the external force is zero then a=0, because the second law refers to all forces, whether internal or external. It is the third law that says the center-of-mass is accelerated only by external forces, i.e., it establishes a meaningful difference between what is "external" and what is "internal" to a system-- forces that come in pairs are internal, forces that don't are external, that's the third law (when you make the simple extension that any system can be made arbitrarily large and thereby include all influences on that system). It's also crucial, because without a distinction between what is internal and what is external, we lose the powerful device of the "isolated system" and its conservation laws. The third law is thus tantamount to the recognition of the existence and importance of an (effectively) isolated system.

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There were other theories (Aristotle) that claimed that motion in a straight line at a constant speed required a constant force.
A fascinating bit of trivia is that Aristotelian physics did not refer to the thing that moved an object forward as a "force" - that's a word that came to have it's modern meaning during the Newtonian paradigm. The word that was historically used was "momentum": they thought an object at rest stayed at rest, and the only reason an object would ever move was because it was acted on by an external momentum. This was divided into two categories: natural momentum due to the four elements, like objects made of Earth having a downward momentum which makes them false, and violent momentum due to the random swerving of objects which is what allows humans to move around.

Ken G
Gold Member
It sounds like that for Aristotle, the word "momentum" meant something close to "intention to move", whether conscious intention or the intentions of the natural order. That was the big disappointment-- Newton showed that a much more useful way to think about motion was that only the changes are "intentioned". That left a level of unintentioned (inertial) behavior that the ancients would have found quite distressing indeed, and probably could not have accepted without a radical change in world view that we take more or less for granted today.