Here’s a question that generated a lot of interest among the people involved in my metals and superconductors short course.

The question is this: Is Newton’s second law – that the net force on a body is equal to its mass times its acceleration – simply a mathematical definition of what a force is? Or is there some additional physical content? If so, what precisely is that content?

Some people regard the answer to this question as “obvious”. As is often the case with interesting questions, one person’s “obvious” may be another person’s “wrong”, so I’d be interested to hear other people’s opinions. If I have time (very busy the past couple of weeks) I’ll put together my own best understanding of the answer.

From → General

Here’s Newton. Principia, definition 8, last paragraph (in english, A. Motte translation):

“I likewise call attractions and impulses, in the same sense, accelerative, and motive; and use the words attraction, impulse, or propensity of any sort towards a centre, promiscuously, and indifferently, one for another; considering those forces not physically, but mathematically: wherefore the reader is not to imagine that by those words I anywhere take upon me to define the kind, or the manner of any action, the causes or the physical reason thereof, or that I attribute forces, in a true and physical sense, to certain centres (which are only mathematical points); when at any time I happen to speak of centres at attracting or as endued with attractive powers.”

Newton’s rules of reasoning in philosophy at the head of Book III of Principia are also worth re-reading. As probably would be what Newton ACTUALLY wrote, instead of vague and over-simple “Laws” that are commonly attributed to Newton in an imprecise form.

It’s very frustrating that so many would rather cut corners reading crib notes on Newton instead of taking the time to read Newtons writings for themselves. There’s really no legitimate substitute for the original works (Principia… and Opticks), certainly not if ones goal is a sincere and educated understanding of experimental science (or experimental philosophy, as Newton put it). If we’re to debate Newton, let us do so with enough respect for the man not to garble and misrepresent what he himself expressly claimed.

Does it matter if one persons “obvious” is another persons “wrong” in light of the content of the text itself? Surely we’re not be take the postmodernists or semioticians tract, and claim Newton didn’t ‘mean’ what he wrote?

2. Fun question. As to Alexander’s comment, I do think it is possible to disagree with Newton Himself (capital H.) I agree that reading the original is interesting, but reading Einstein’s original manuscripts on general relativity would lead you to conclude the universe is static! To suggest that Newton understood fully the consequences of what he was doing is a huge leap which I am not willing to make.

The “it’s only a mathematical definition of a force” point of view says force is simply the thing we use along with mass to calculate the acceleration. But I think the 2nd law says a lot more than that. First of all there is that funny thing, the mass. If we subject different masses to the same set of circumstance which produce a force, then we get different accelerations. Of course it didn’t have to be this way. There could not be something called mass: depending on what type of force is being applied the acceleration corresponds to the second law with a different mass, for example. (In fact this is the reason why you could define different types of masses, gravitational mass, electromagnetic mass, etc. There is no reason these should be related to each other!) More interestingly, why isn’t the mass a tensor? If this tensor is not an invertible matrix, then there is now no way to “move it into the definition of the force” in a nice way. Indeed this would be a strange world. So I’d call this some physical content.

The second interesting part of the equation, is, of course the force. Here the interesting thing is the fact that it is a vector quantity. Again, it didn’t have to be this way, Why when we have two sorces of force should we add them together as vectors? Why not take their cross product (what a funny world that would be!) The world appears to be set up so that we can define forces and then add them as vectors (very strange…especially if you were Plato or Socrates!) I’d call this a physical content of the second law as well.

As you can see, my definition of physical content is something like “that which tells you the world is one way when it could have been all of these other ways.” Sure the equation defines the terms it uses (“To speak is to fall into tautology…you who read me, are you sure you understand me?”) But it does more than this: if the law is upheld it limits the world in a particular way. And in this case a particularly beautiful and interesting way.

Just my 2 cents. I hope someone disagrees strongly so I can more clearly see the its just a mathematical definition point of view!

(My free time to write this sponsered by a complete reinstall of the hard drive I nuked.)

The physical significance of Newton’s second law is explained in the fifth paragraph of Landau and Lifshitz (translated by Sykes and Bell):

“If all the co-ordinates and velocities are simultaneously specified, it is known from experience that the state of the system is completely determined and that its subsequent motion can, in principle, be calculated. Mathematically, this means that, if all the co-ordinates q and velocities qdot are given at some instant, the accelerations qdotdot at that instant are uniquely defined.”

In other words, equations of motion are second order DEs. It is implicit in Newton’s second law that forces depend only on the configuration of the system and its first derivative with respect to time.