# F=ma: definition? law? neither?

1. Apr 18, 2006

### Nancarrow

Hi all, sorry if this has been done to death in schools everywhere for three centuries, but I haven't seen any textbook that really answers it for me.

I understand Newtonian mechanics, at least at the level of solving exam questions. But ever since I decided to go back and relearn everything from scratch ('second quantisation' did it for me ), I've been stuck at the first hurdle.

In school the teachers told us it was important to distinguish between equations that just defined quantities, and equations that described the physics of the world. But I can't understand where Newton II fits into this. Everywhere I've looked describes it as a 'law', but AFAI can see it's a definition of force (as the rate of change of momentum).

I've heard it described as 'half a law'. As in, you write it down for a body, then you plug stuff in, to replace the 'F', according to gravity, friction, electrostatics or whatever the problem at hand is. But 'half a law'? Don't like the sound of that.

What I think so far is, NII is simply a definition of force, and the actual physics is in the law of momentum conservation.

What say you, knowledgable people?

What's the definitional content of Newton Mech, and what's the physical content?

[disclosure: I have a physics degree. As you may guess, I didn't do very well ]

2. Apr 18, 2006

### Hootenanny

Staff Emeritus
Firstly, Newton's second law is not F = ma, Newton's second law is (as far as I know);

"Mutationem motus proportionalem esse vi motrici impressae, et fieri secundum lineam rectam qua vis illa imprimitur."

Which translates to "The rate of change of momentum of a body is equal to the resultant force acting on the body and is in the same direction."

$$F = \frac{dp}{dt}$$

I'm not quite sure what you're asking, if you could please clarify.

Regards,
~Hoot

3. Apr 18, 2006

### Nancarrow

Yeah, I just chose 'F=ma' as it's more immediately recognisable. Of course F=dp/dt is the more correct form of the law, F=ma only holding for constant mass.

I'm not surprised you don't understand what I'm asking, since I keep wondering if I'm just chasing my own tail here.

OK: So, F=dp/dt is a "law"? If so, then it is stating a fact about the motion of bodies in the physical universe. It is placing a constraint upon the motion of an object, out of all the possible paths that it could take, the 'laws of physics' must tell us which one it actually does take. F=dp/dt does not do this, as it stands. You've got to replace F with something. Say, 'mg' for a simple projectile problem, or 'GmM/r^2' for an orbit problem.

Here's what I'm getting at: if F=dp/dt is a law, what is the definition of a force?

4. Apr 18, 2006

### vanesch

Staff Emeritus
You are right, this is sometimes confusing. The answer depends upon the point of view of the author. In fact, Feynman, in his lectures, adresses the issue. I also remember getting confused by reading about the DEFINITION of a force acting on a body as being the mass of the body times its acceleration, and then hearing that this was a law.

Feynman illustrates the issue by "defining" the concept of a "gorce". The gorce acting on a body is the mass times its velocity ; and the total gorce acting on a body equals the vectorial sum of all gorces acting upon it of other bodies. Thinkable. Totally useless.

So, yes, you can take F = m.a as the definition of force (in the same way that G = m.v is the definition of a gorce). And then you have to find "laws" that make you able to calculate F from the particular situation. It is the *feasability* of this programme for the force that makes it a *useful* definition. It's the fact that we CAN say that F = -G m1 m2 1_r / r^2, and that this is the same as what was defined to be the force acting on a body through F = m.a, that makes that the definition of force has a useful meaning (and not, the definition of a gorce).

You can have then the trivial law of conservation of position: "any body on which no gorce acts, has constant position". It's true.
This is the equivalent of the law of conservation of momentum: "any body on which no force acts, has constant momentum". It's true too.

But the difference is that the last situation corresponds to "a body which is free of influences from other bodies", while the first situation doesn't correspond to a simple situation at all.

Therefor the concept of force, though it is DEFINED as being the change in momentum, is intuitively actually understood as *interaction* between different systems (the sun's gravity force pulls on the earth ; without sun, no such gravity force - though of course a gorce).

That's why there are authors who do NOT consider F = m. a as the definition of a force, but as something intrinsic about the interaction between bodies. F, acting on a body, is then the vectorial sum, by definition, of all the forces through interaction that act on a body. Now, you'd have then to define the force of gravity, the force of electromagnetic interaction...
and the LAW is now that if you sum all of these forces (through their definitions), that then, you find out that this sum equals m.a in inertial frames. But this approach ALSO has something strange about it, namely, why should we DEFINE the gravity force as G m1.m2/r^2 and not as, say, G' m1^4/m2^2 / r^2 or the like. HERE ALSO, what is proposed to be an arbitrary definition is something which is highly inspired by the actual behaviour of nature.

So there are the two approaches:

1) definition: F = m.a
law: for gravity, F = - G.m1.m2/r^2

2) definition: F = - G.m1.m2 /r^2 for gravity
law: F = m.a

but NEITHER is an *arbitrary* definition. It's because this system WORKS, that we define it this way. We didn't make it work starting from an arbitrary definition. So both are "laws" of nature in this sense, because they both derive their sense and usefulness from the fact that it works, this way, in nature.

5. Apr 18, 2006

### ZapperZ

Staff Emeritus
If you have access to Physics Today, I strongly suggest you read a tour de force 3-part article written by Frank Wilczek on F = ma.

PHYSICS TODAY, October 2004, page 11, December 2004, page 10, and July 2005, page 10.

Zz.

6. Apr 18, 2006

### Andrew Mason

An excellent article that can be found here. A veritable tour de force or as Wilczek might prefer, a tour de gradient potentiel.

F(=AM)

7. Apr 19, 2006

### Nancarrow

Thanks for your long and detailed reply Vanesch! I think it's helped. A bit! I've got the Feynman lectures, I'll have a look this evening. FTR I never meant to imply that the definition of force was in any way 'arbitrary', of course it's extremely useful. I was just hung up on the conceptual difference between definition and law.

One thing that I think I'm getting from all this, is that the law of conservation of momentum for *systems* of particles (which = Newton's third law, integrated over time, right?), really *is* a law and not just a matter of definition. NII can be taken as a definition of force, provided that further laws are given, stating what forces actually occur and under what conditions (eg gravity, collisions...)

I'll also definitely check out that Wilczek article, thanks for the pointers, ZapperZ and Andrew Mason.

'Tour de force'... har :rofl:

8. Apr 19, 2006

### Andrew Mason

Conservation of momentum for isolated systems (ie. those which do not interact with other matter) is simply a consequence of:

$$\frac{dp}{dt} = 0$$

If $\frac{dp}{dt} = x \ne 0$ we have defined a quantity, x, that measures the extent to which momentum is not conserved, which is simply a measure of the magnitude of the matter-matter interaction. Of course, this is the definition of 'Force'. The 'law' part of it could be considered as simply: the magnitude of system's interaction with external matter determines the rate of change of its momentum.

AM

9. Apr 19, 2006

### vanesch

Staff Emeritus
Well, the "law" is in the last part: that this non-conservation of momentum has something to do with "interaction with other stuff", and is nothing "inherent" to motion. And then, even THIS is more subtle than one could think at first sight, because, in order to even be able to SAY
$$\frac{dp}{dt} = 0$$
one has to say in WHICH REFERENCE FRAME one does it. Clearly, if I'm shaking up and down and use myself as defining a reference frame, then there's not much stuff (apart from things attached to my body, such as my glasses) that obeys this law. Certainly not objects of which we could assume that they are free of interaction, or almost so.
And if one uses "objects free of interaction" to DEFINE the reference frames in which this law/definition/whatever $$\frac{dp}{dt} = 0$$ is to be written down, then one should be extremely careful of circular reasoning!
So all this is a complicated mesh (mess ?) of hidden assumptions, definitions, observed phenomena which is very difficult to dissect into "definitions" and "observed laws" (or principles or whatever).

So it seems we have to start with the definition of what is a (general) reference frame. And even *that* is subtle! You have to map in some way, points of an Euclidean 3D space onto "points in physical space", and you have to define a time (a clock). How ? Randomly ? Well, do it with meter sticks (and *postulate* things about how meter sticks behave when displaced) and a specific clock. What's a clock ? Subtle issue by itself ! A Rolex of course
Now consider that there can be observers, essentially "free of interaction", and who apply this meter-stick + clock procedure to set up their reference frame. But they can still be rotating ! So define an entire family of these observers, all rotating with respect to eachother. Fine. Now, there is a LAW OF NATURE that says that ONE of these observers (at least), in his thus-defined coordinate system, with his clock thus defined, will observe the motion (coordinates vs clock readings) of OTHER interaction-free objects (you postulate that those exist, and how to recognize them) to be straight lines. THIS observer is then an inertial observer, by definition, and you say that it is a LAW OF NATURE, that at least one inertial observer exists.
Next, you need to say how different inertial observers will relate identical events. This is done, in Newtonian physics, through the Galilean transformation (x' = x - vt, y'=y, z'=z, t' = t in its simplest form). This, again, is a LAW OF NATURE.
The existence of an inertial observer, plus the law of Galilean transformations, plus the definition of inertial observer, then leads you to the statement that all inertial observers have constant velocity wrt eachother.
And NOW, we're ready to say: $$\frac{dp}{dt} = 0$$
for objects which are free of interaction.

10. Apr 19, 2006

### Andrew Mason

All very good points. It was implicit in Newton's analysis that there existed a frame of reference that was not acted on by a net force. A fixed point on the earth resembles an inertial reference frame but, of course, it is not. In fact, there are no truly inertial reference frames; merely frames in which, for the purpose of doing certain experiments, one can say that external forces are sufficiently small that they can be ignored.

The point I wanted to make is that F=dp/dt - in fact all three of Newton's laws - become laws if one defines force as the 'magnitude of the interaction between two objects or systems of matter'.

1. The first law is: For a matter object, dp/dt = 0 if there is no interaction of the object with other matter.

2. The second law is: For a matter object, dp/dt = F where F is the magnitude of the interaction of the object with other matter.

3. The third law is: For any system of matter objects, $$d(\Sigma_{i=1}^n p_i)/dt = 0$$
if there is no interaction between the system and other matter.

AM

11. Apr 19, 2006

### masudr

I always thought the first law defined the arena of Newtonian physics: inertial reference frames.

12. Apr 19, 2006

### pmb_phy

Please excuse me if I repeat some of the above but my back won't allow me to sit here and read the whole thread up to this point. I read some and scanned the rest.

Force is defined as dp/dt. When it comes to gravity we have dp/dt = GMm/r2.

There is a good article in the American Journal of physics on this. See

On force and the inertial frame,Robert W. Brehme, Am. J. Phys. 53, 952 (1985)

There is also an entire book on the concept of force. It was written by the well-known physicist Max Jammer and its a cheap Dover book. I recommend it highly. After you read the book I doubt that you'll have any doubts on what force is. He has a book on mass and on space too.

Pete

13. Apr 20, 2006

### vanesch

Staff Emeritus
Indeed, that's what I said in one of the previous posts:
Honestly, I also prefer that approach, and it is what you intuitively THINK of when you think of "force": *something* is pulling or pushing, hence interaction.

However - and it confused me too when I first read it - often force is DEFINED as dp/dt. I started learning "serious physics" on my own when I was in high school using Alonso and Finn. They take on this approach of defining force as the change in momentum and it seriously confused me as I remember ; only to understand the issue (I think ) when I read the Feynman lectures a few years later.
If you DEFINE force as dp/dt, it has become a purely kinematic concept ! It has nothing to do with interaction - which is not what it was meant to be, in the first place. It is maybe the easiest formal way of getting things going, but it certainly is not the *idea* behind the concept of "force", which is, as you suggest, more a notion about interaction than about kinematics.

EDIT: BTW, this is what was so brilliant in Feynman's "gorce" approach, and illustrates exactly all the issues that come into play with Newton's laws and the associated definitions.
Remember that Feynman defined G = d/dt (m x) to be the total gorce acting upon an object, and consider what it means to have "conservation of m x" and so on. So you could define an inertial observer as one for which objects on which no Gorce acts, to obey the above law (meaning: they're standing still). This is a more severe requirement of inertial observer: in fact, instead of having a 10-dim galilean group, we have a 7-dim group (translations in space, rotations in space, and translations in time, but no boosts: in other words, an absolute Euclidean space + a time axis). In fact, this view is not totally ridiculous: it is close to the Aristotelian view on nature - objects move until they find their natural place ; and this "desire to move" is nothing else but gorce... Although of course no classical philosopher wrote down such an equation.

While all this is a perfectly good scheme, what is clearly missing is the relationship between G and "interactions with something else" and a way to quantify this.

In the same way, one could play a game with J = d/dt (m a), the total jerk acting on an object.

Gorce, Force and Jerk all have a well-defined kinematical meaning, but only Force has something to do with interactions, and hence has a dynamical meaning.

Last edited: Apr 20, 2006
14. Apr 20, 2006

### Nancarrow

Hi again, I'm still around, I'm just trying to digest everything everyone is saying, and have nothing useful to offer at this point. Except to say that if Feynman, Wilczek and you lot can spin this out into long discussions, it makes me feel like less of a twit! :tongue:

ETA: I'm reading Chandrasekhar's edition and commentary of the Principia at the mo, and he quotes extensively from Maxwell's treatise on mechanics, so that's two more bigwigs I can rub shoulders with. I feel almost clever now.

15. Apr 20, 2006

### JoAuSc

After thinking a while back about what Feynman said about gorces, I realized that what F = ma implies is that the universe is such that (for the most part) acceleration is a function of position. y" = f(y), where y is the position vector and y" is d2y/dt2. For gravity, it'd be something like f(y) = 1/m * GmM/r^2 in the direction of the other particle.

It's interesting that you mention that. I've been wondering for a while what the universe would be like if instead of y" = f(y), we had y''' = f(y) or y' = f(y) or some other differential equation.

16. Apr 20, 2006

### masudr

Funny that; me too!

17. Apr 21, 2006

### vanesch

Staff Emeritus
Well, almost. Magnetic forces, for instance, aren't function of position, but of velocity and position. But the thing to realise is that this "position" and this "velocity" that enter into the expression for acceleration, are entirely function of the "position" and "velocity" of OTHER THINGS AROUND.

The point is that acceleration of thing A seems to be function of "the dynamical state" of the other things around as compared to the dynamical state of A, and that those influences add vectorially. This is a good indication that those influences are "interactions". It is because of the presence of those other things that our thing A suffers an interaction, and these interactions seem to be independent of one another (in that we can calculate the effect of the presence of each thing individually, and then SUM all these effects vectorially). We then call such an "individual effect of another thing upon our thing" the FORCE excerted of this other thing on our thing, and we now have the fact that the forces, which are the effects of thing B, thing C and thing D on our thing A, add vectorially together to make up the *total force* acting upon A.
F = F_B + F_C + F_D, and F_B is ONLY a function of the relative dynamical state of A and B, F_C is ONLY a function of the relative dynamical state of A and C, and F_D is ONLY a function of the relative dynamical state of A and D.

And why is a dynamical state defined by position and velocity ? Well, because inertial frames are! It is because - remember the LAW OF NATURE that specified an inertial frame - things, free of interaction, trace out a straight line with uniform motion. The degrees of freedom here are position and velocity (the initial conditions you can freely choose and have different possible states of motion of things free of interaction).

So the dynamical state of a thing, free of interactions, is given by its position and velocity. That makes it plausible to say that this remains so for a thing with interactions, so if interactions are to be just that, they should ONLY depend upon the dynamical states of the two things interacting. And because of the principle of Galilean invariance (the free choice of an inertial frame), you can always look upon things from the instantaneous tangent reference frame going with the thing A, and hence the interaction, in this frame, on A, from a thing B, can only depend upon the dynamical state (position and velocity) of thing B ; in other words, the relative dynamical state of B and A.

But, but, what's entirely NEW - again a law of nature, which I don't think has any specific name - is, that it seems that interactions come one by one and are just between TWO bodies. That there is no intrinsic 3-body interaction. The Newtonian framework doesn't forbid this, but it doesn't seem to be the case. Interactions go 2 by 2, and seem to be vectorially added independently.

18. Apr 23, 2006

### cliowa

Actually you don't need to have access to Physics Today in order to view these articles: Just go to Frank Wilczek's MIT Homepage: http://web.mit.edu/physics/facultyandstaff/faculty/frank_wilczek.html There's a list of his publications where you can select "Publications in Physics Today" and you'll get all of'em as .pdf-files.