# I don't understand the ''definition/law'' aspect of Newton's laws

1. Jun 9, 2014

### Coffee_

Hello everyone, it's been a while since I've been introduced to Newton's laws for the first time. Yesterday I've been reading the Feynman lectures vol 1, where Feynman makes a remark about Newton's laws not really containing any physics. This got me thinking and I'm really confused right now. English is not my first language so I'll do my best to get my confusion along to you. I'm talking about the first two laws only:

The problem arises when I start looking at Newtons first law without any pre conceived idea of what a force is. I have read that this is one way of approaching them. So let me look at the following statement:

''In inertial reference frames, the velocity vector of an object remains constant only and only if no force is acting on it.''

Two new concepts introduced: A) Inertial reference frames B) Force. To counter pre conceived notions sneaking into my reasoning I'll rename the things as A) Special frame B) Wobble.

So basically I have ''In special frames, the velocity vector of an object remains constant if there is no Wobble''

As you see this doesn't make sense. Let's say I'm looking at an object A moving with a constant velocity vector. If I knew that I was in a special frame I could conclude that there was no Wobble. However, I can't know if I'm in a special frame or not because they were not defined before this moment. Other way, if I knew that there was no Wobble I could say that I'm in a special frame, but again I can't know if there is a Wobble or not.

So the first law doesn't introduce anything, even not a definition. It could be viewed as:

1) The definition of a special frame in function of this other unknown thing 'Wobble'

2) The definition of a Wobble, using another unknown thing ''special frame''.

2. Jun 9, 2014

### dEdt

Your basic point is correct, namely that it's impossible to use the first law to define both inertial frames and force. But force can be given a definition independent of the first law. Roughly speaking, force is the influence of one object on the motion of another. It's not really a precise definition, and there's certainly some room for ambiguity, but we're dealing with science, not mathematics.

So now that we're equipped with a definition for the term 'force', we can use Newton's first law to define inertial reference frames.

3. Jun 9, 2014

### DrStupid

That makes no sense. Force and inertial frames are defined by all three axioms.

4. Jun 9, 2014

### Coffee_

So basically, the way to know that there is no force on an object, is to put it away from any possible influences. Once I know that there is no force I have an unambiguous statement how to find if I'm in an inertial frame or not. If I find myself to be in an inertial frame, I make another definition:

''F=dp/dt in inertial frames.'' This one strengthens the earlier vague definition of what a force was right?

Am I correct to say that Newtons second laws introduces no ideas but really is a definition. It was the first law that introduced the new idea of ''no acceleration iff no force'' for inertial reference frames. The second law defines a quantitative relationship.

5. Jun 9, 2014

### Philip Wood

Yes, and an inertial frame is one where all the forces on a body can be attributed to the influence off other bodies. So, for example, in a rotating frame there are centrifugal forces, but these cannot be attributed convincingly to the influence of other objects.

6. Jun 9, 2014

### Coffee_

I think I see, basically my reasoning mistake here was to try and understand Newton's laws without a pre conceived idea of what a force means right?

7. Jun 9, 2014

### D H

Staff Emeritus
The modern view of Newton's laws is that the first law of motion is definitional, that it defines the concept of an inertial frame of reference.

The second and third laws to me truly are "laws" (better: physical theories) rather than merely definitional. The second law addresses the linear relation between net force, mass, and acceleration. Note: F=ma is the modern view of Newton's second law. Newton used a lot of circumlocution to avoid using his calculus in his Principia. And even if he had used calculus (and vectors, which are 200 years post-Newton), he would have written his second law as F=kma, where k is some constant of proportionality. He and others before him (most notably Galileo) tested for the linearity. It is not definitional. The third law also is a testable physical theory.

Both the second and third law don't hold up in light of Maxwell's equations, relativity, and quantum mechanics. They are falsifiable statement (what else could you ask of science) and indeed have been falsified. So why do we still use them? Popper's theory of falsification itself is falsifiable, and it too has falsified. We still use Newtonian mechanics because it is correct to within measurement errors in a limited regime where velocities and masses are small and distances are large. ("Small" velocities means small compared to the speed of light. "Small" masses means compared to, for example, the black hole at the center of our galaxy. "Large" distances means compared to the distances between atoms in a crystal or to the Schwarzschild radius of some object). That limited regime is the regime of our everyday experiences.

There are two things missing here:
- What are forces?
- What's this net force business in the second law versus the individual forces in the third?

Newton addresses the second item in his corollaries to his laws of motion. The modern view of those corollaries is that forces are subject to the superposition principle, i.e., that forces are vectorial. Newton's proofs of those corollaries are a bit lacking to me. He appears to be begging the question in those proofs, assuming what he is trying to prove. In fact, the modern view is to make those corollaries axiomatic by telling students that forces are vectors.

But what are forces? Newton addressed mass, momentum, distance, time, and force up front, before he stated his laws. They're undefined terms. There's nothing wrong per se with undefined terms. After all, Euclid never defined what a point, a line, or a plane are. Those three things are the undefined terms in Euclidean geometry. Mass, momentum, distance, time, and force are essentially the undefined terms in Newtonian mechanics.

8. Jun 9, 2014

### sophiecentaur

Don't we just need to come to terms that any of these quantities are merely constructs of our minds / culture that allow us to make predictions and justify / explain / rationalise our observations? The question "what is" is just as suspect as the familiar "why" question.
Force is a concept we apply in mechanical situations. It works for me and for you. It was not used in ancient history but they still managed to make their things work. One day, humans may look back on Force in the same amused way that we look upon Miasma or 'Nature abhors a vacuum'

9. Jun 9, 2014

### DrStupid

To me all three laws are definitional.

The modern view (and most likely also the original intention) of Newton's second law is F=dp/dt. Going from m·a to dp/dt makes no sense because the former is easy to understand whereas differential equations where unknown that time except for Newton himself. But his formulation makes sense if he started from conservation of momentum and didn't made the step from dp/dt to m·a. Today we know that this step would have make the axioms invalid in modern physics.

I do not see a chance to test them.

This is answered by Newton's laws of motion:
Lex 1: Forces are the cause of acceleration.
Lex 2: Forces are proportional (in modern view equal) to the change of motion.
Lex 3: Forces cancel each other pairwise out.

Of course it would have been sufficient to define that the net force need to be zero. But pairwise forces that cancel each other out are sometimes easier to handle.

Momentum is defined in Definition 2. Force is defined by the the laws of motion. Quantity of matter (we better do not mix the term "mass" with Newton because he wasn't aware of the modern convention regarding its use) is defined by Definition 1 but this definition is not useful. Fortunately Definition 2, Lex 2, Lex 3, isotropy (mentioned in the introduction of the Principia) and the transformation between inertial systems provide an implicit definition of this property. Distance and time seem to be assumed as known.

10. Jun 9, 2014

### dEdt

For what it's worth, here's my perspective on these things:

As I mentioned earlier, before we discuss Newton's laws we need a qualitative definition of the word 'force'; specifically, we'll define force as the influence of one object on another's motion. If you're happy with this definition, then we can move on.

The first law states that there exists a frame of reference such that all force-free objects move with constant velocity. Such reference frames are called 'inertial reference frames'. Note that while the first law defines inertial reference frames, it is not merely definitional because it is a highly non-trivial claim that such reference frames exist in the first place.

The second law is a bit trickier. It is usually stated as "in an inertial reference frame, $\mathbf{F}=m\mathbf{a}$ ." But all this is is a mathematical definition of force, and it's not clear how a mere definition can be the bedrock of classical physics.

What's left unstated in the usual formulation of Newton's second law is that this definition of force leads to simple formulas. More precisely, any other conceivable definition for force will in general yield more complicated equations. For example, the force between two charged objects is $\propto q_1 q_2 / r^2$ using this definition of force. If you used some other definition, the formula for Coulomb force would be much more involved. This, then, is the physical content of the second law: in a general dynamical system, the value of $m\mathbf{a}$ for any given particle will take on a surprisingly simple form, making it the natural definition for the force on that particle.

I would also include the superposition principle in the second law. Indeed, the fact that force as defined by Newton obeys the superposition principle is another demonstration of the appropriateness of his definition.

I think the third law is fairly self-explanatory.

11. Jun 9, 2014

### dauto

Yes, Newtons laws by themselves have almost no content. They only bloom when laws of forces are added to the mix such as the law of gravity for instance.

12. Jun 10, 2014

### D H

Staff Emeritus
Sure they do. Newton's third law (weak form) leads to conservation of momentum. The strong form adds conservation of angular momentum. Conservation of energy results when forces can be expressed the gradient of a scalar function of position.

Another way to look at it: Lagrangian and Hamiltonian physics are just Newtonian mechanics recast in a more elegant form.

13. Jun 10, 2014

### Jano L.

Newton's laws as he stated them are not definitions, but positive statements about the world. People often think some of them are definitional because they use the term "Newton's laws" for completely different statements from modern mechanics, which were unknown to Newton. These new statements, using concepts of inertial mass, inertial frame, acceleration, vectors or other anachronisms resemble Newton's laws but are not the same thing. We should make clear what we are talking about when we are talking about Newton's three laws.

The original statement of Newton's three laws:

LAW I.
Every body perseveres in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed thereon.

LAW II.
The alteration of motion is ever proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed.

LAW III.
To every action there is always opposed an equal reaction : or the mutual actions of two bodies upon each other are always equal, and directed to contrary parts.

It should be clear that these statements are not mere definitions.

One modern set of statements resembling/related to Newton's three laws:

I.
There exist inertial reference frames.

II.
Force acting on a body is a vector quantity defined by $m\mathbf a$ where $m$ is mass of that body and $\mathbf a$ is its acceleration.

III.
Any force acting on a body due to another body is accompanied by force on the second body due to first body of the same magnitude but opposite direction.

Of these statements, II. is clearly a definition. But it should be clear that first two statements are very different from first two statements of Newton's laws.

There are many other similar triples of statements related to Newton's three laws, with number of definitional statements anywhere from 0 to 3.

Neither Newton's laws nor their reformulations of the above kind are sufficient for understanding the laws of mechanics. Other information is necessary, either in the form of definitions or other laws.

14. Jun 10, 2014

### sophiecentaur

I don't understand the ''definition/law'' aspect of Newton's laws.

Newton's ( and anyone else's) 'Laws' are only, ever, enough to make reasonable predictions. They are not, in themselves, 'definitions' but they all hang on definitions that are, somewhere back along the line, based on assumptions and gut feelings.
Those pesky Philosophers do their best to regularise this situation but they end up applying logic, which is not necessarily a part of the Physical world. Their stuff is not falsifiable in the way that Science tries to be.

15. Jun 10, 2014

### Coffee_

Why? I can always say that if the object undergoes only translational motion that I set my origin of my reference frame in the mass center of the object. Whatever the motion of an object is I can always figure out what reference frame where v=0, even if this means to let my frame accelerate with the object. This conclusion uses only more or less math reasoning and no assumptions about the physical world.

16. Jun 10, 2014

### Jano L.

Such reference frame is not always inertial, because test particles around the object will not move rectilinearly in its reference frame.

17. Jun 10, 2014

### stevendaryl

Staff Emeritus
I think of Newton's laws as more of a framework for physics than a particular theory of physics. If you don't know what forces are at work in a particular situation, you can't apply $F = ma$ in a way that has any predictive value (you could calculate F by measuring m and a, but that doesn't allow you to predict $a$). On the other hand, if you supplement Newton's laws with specific hypotheses as to the types of forces at work, then Newton's laws plus force laws together make a predictive theory. If you assume that a spring stretches when a force is applied according to Hooke's law: $F = -k \delta x$, then forces become measurable independently of accelerations, and then $F = ma$ is a testable claim.

Without making some kind of assumption about the forces at work in a particular situation, I don't see how you can get anywhere using Newton's laws of motion. Newton's third law sounds like it is falsifiable, because you can check to see if the acceleration of one object is accompanied by equal and opposite changes of momenta of other objects. But if you don't know what forces are at work, then you don't know how far away to look for these other objects.

Another problem with Newton's laws is mass. We have no way of measuring the mass of something other than checking how it moves under the application of a known force.

In practice, science is not as clean as the naive interpretation of Karl Popper's falsifiability standard would imply. You have to make certain auxiliary assumptions about mass, forces, etc., in order even to make sense of your measurements. Only then can you use measurements to test your theories about forces and masses. So it's really an iterative process: You make some initial assumptions. You make measurements based on those assumptions. You use the measurements to test your assumptions. Then you may need to make new assumptions based on those tests, which may require revising your measurements. Hopefully, the process converges.

18. Jun 10, 2014

### dEdt

You can always find a reference frame such that one force-free object moves with constant velocity, no matter what the laws of physics are. But for all we know, there could be another force-free object moving in circles in this reference frame. What's non-trivial is that all force-free objects move with constant velocity.

19. Jun 10, 2014

### DrStupid

Why? All three statements together provide a qualitative and quantitative definition of force:

The first law is a qualitative definition (force is the cause for alteration of motion).
The second law is a quantitative definition (force is proportional to the alteration of momentum).
The third law distinguishes forces from fictitious forces (forces have corresponding counterforces).

The definition of inertial systems is just a side-effect effect: Inertial systems are characterized by the absence of fictitious forces.

20. Jun 10, 2014

### DrStupid

That's what they are intended for. In Newton's Principia the laws of motion are the basis for the law of gravitation. The laws of motion define what force is and how it affects the motion of bodies and the law of gravitation describes how force depends on mass and distance of the bodies. The laws of motion would be useless without the law of gravitation and other empirical laws describing (and predicting) forces under specific conditions and these empirical laws would be meaningless without the laws of motion.

Most scales measure the mass of bodies at rest. Using inertia to measure the mass is a rather exotic method.