By Michael Nielsen, November 23, 2019

Note: Rough working notes, me thinking out loud. Thoughtful, well-informed further ideas and corrections welcome.

The laws of physics don’t need justification, they just are. In that sense, “why does F = ma?” is a ridiculous question. Certainly, it can’t be proved, it’s not a mathematical theorem, or in any sense inevitable. Indeed, it’s easy to imagine universes in which F = ma is not true: we live in such a universe, since F = ma only arises as an approximation to a deeper quantum mechanical reality.

Nonetheless, there’s a sense in which “why does F = ma?” is a stimulating question. It’s a challenge to deepen one’s understanding of Newton’s second law of motion, and to understand how the universe would be different if we replaced the second law by something else.

As a student, Newton’s second law bugged me. Physicists often seemed to use it almost tautologically, as a definition of what a force was. I wondered if there was any non-tautological, non-trivial physical content? Or was it really just a definition? Of course, it turns out there is deep physical content, but it’s left implicit in many discussions of Newton’s second law. Let’s make it more explicit.

(Incidentally, the equation in Newton’s second law isn’t F = ma, but rather the more subtle statement that force is equal to the rate of change of momentum of a body. When the mass isn’t changing, that reduces to F = ma, and that’s the form I’ll discuss here.)

One way to get insight into the second law is to consider variations. How would the world be different if instead of F = ma, we instead had F = mv, that is, force is equal to mass times velocity? Or perhaps F = mj, where j is the “jerk”, the third derivative of position. Is there some reason Newton’s law involves the second time derivative of position, rather than the first or the third (or some more exotic) derivative?

There’s a conventional answer to this question. The key to this answer is that if we take some fixed configuration of (say) gravitating bodies, and then consider a test particle, its subsequent motion: (a) is completely determined by its initial position and velocity; but (b) the initial position and velocity are free variables which can be changed relatively easily.

So, for instance, you can’t have F = mv, because that would mean the initial velocity would be entirely determined by the configuration of surrounding matter. It would actually be impossible(!) for us to set the velocity of (for instance) a projectile. But in practice we find that initial velocities are things which we have a lot of freedom to adjust. So F = mv is ruled out.

In more mathematical terms: suppose we believe the motion of a test particle is completely determined by its initial position and velocity, but also that those quantities are free variables which we can choose. If we know just a little about differential equations this suggests some kind of second-order differential equation must be controlling the behavior of the particle. In particular, the acceleration of the test particle should somehow be a function of the other configuration of matter. F = ma is very nearly the simplest equation we can imagine of this form; the mass is the only slightly unexpected feature in the equation.

This is a pretty conventional story. It’s one I remember reading in textbooks as a student. It has some insight worth remembering, but it’s wrong in important ways. For one thing, test particles don’t all behave in the same way. Two test particles with the same initial position and velocity, but different electric charges, can behave quite differently in the same electric field.

One possible response is to say “oh, maybe our notion of force should really be something like F = mj, where j is the jerk, i.e., the third derivative of position”.

I’ve never worked it out in detail, but wouldn’t be surprised if such an approach can be made to work. Essentially, it’d make acceleration into a free (possibly constrained) parameter of the particle, rather than something completely determined by the distribution of matter and fields. That free parameter would implicitly contain what (in the conventional approach) we think of as the charge information. Indeed, the new equations of motion would have a conserved quantity, corresponding to the charge. But the resulting force laws would be quite a bit uglier.

(Actually, if we ever saw a situation in nature where charges seemed to change over time, this jerk-based approach might be worth exploring!)

So what then really is the content of Newton’s second law?

The right-hand side of F = ma is at least moderately clear, though it bears more examination.

But the left-hand side, the very notion of a force, is subtle indeed. There’s an underlying implicit set of assertions: matter produces forces on test particles; those forces control the behavior of the test particles; those forces can be computed as a reasonably simple universal function of the configuration of matter and fields, notably of positions, velocities, and charges.

(Just to make the last assertion more concrete: Newton’s law of gravitation, for instance, asserts that you can compute the force on a test particle as the integral over mass density throughout the universe, in accord with the inverse square law. And, of course, other people have figured out other ways of computing force as a function of the distribution of matter and fields.)

None of these implicit assertions has anything a priori to do with ma. Rather, they’re a remarkable set of assertions about how we should describe nature. And they’re all implicitly part of the content of the second law, though often not so explicitly stated. If these things weren’t true, the second law wouldn’t be a useful statement; indeed, no-one would ever have heard of it.

Putting it in somewhat fuzzier terms, and at the risk of repeating myself: F = ma derives its power from the (implicit) assertion that there is a simple universal force law that lets us figure out F for a particular configuration of matter. And so the configuration of matter completely determines the acceleration of a test particle. There is no a priori reason this ought to be true. It’s an absolutely incredible fact of nature.

Let’s condense our observations into a single paragraph: a reasonable answer to “why does F = ma?” is: the behavior of test particles is somehow determined by a quantity which we’ll call a “force”. This force is a simple function of the configuration of matter and fields, notably of the positions, velocities and charges of all particles. In practice, we find it’s possible to change the initial position and velocity of test particles, without changing the rest of the matter configuration. But it doesn’t seem so easy to change the initial acceleration, without changing the rest of the matter configuration. That suggests the force should somehow determine the acceleration. At this point, F = ma seems a good candidate law of motion.

Personally, I find this all a very helpful line of thought. Of course, there’s still much that’s mysterious. For instance, I haven’t said anything about why m appears in the second law, or even where the notion of mass comes from. Of course, mass is very familiar to us from childhood, and so seems innocuous, but it’s an incredibly deep and subtle idea. What’s it doing in the second law? If F is a universal function, then m is almost like a resistance, something that makes a test particle respond less to the applied force. It’s remarkable this is a fixed constant for particles in nature.

A fun question: how does the universe change if the mass isn’t a scalar, but rather is a matrix, and so a = m-1F is the acceleration? What would this world look like? Is it plausible?

Another fun question: how does the universe change if F = mw, where w is a fractional time derivative of position? Say, for instance, the 1.5th time derivative of position. Is there any sensible formulation of (classical) physics where this kind of thing can be used as a law of motion?

And one more fun question: is there any connection to evolutionary psychology? Human beings can see (and manipulate) the position and velocity of everyday objects quite well; much, much better than they can see the acceleration. People routinely get the sign(!) wrong when estimating acceleration; it’s hard to imagine that happening for velocity, outside of rather contrived circumstances. Is there some evolutionary reason for this, connecting Newton’s law to facts about our nervous system?

Of course, it’s possible to deepen our thinking much further. We can start to think about F = ma as a consequence of the Euler-Lagrange or Hamilton’s equations; or as a consequence of the Schroedinger equation, or of Feynman’s sum-over histories approach to physics. Indeed, I suspect it is possible to in some sense deduce the second law of motion from thermodynamics. (Cf the work by Ted Jacobson on the Einstein field equations as equations of state, and more recent followups). And we can think much more deeply about notions like “test particles”, or what reference frame to measure acceleration in. And so on – a panoply of great questions! Newton’s laws are incredibly deep.

An interesting feature of the discussion above is that it’s written for people already familiar with Newton’s laws. It takes as given a lot of pre-existing intellectual structure. I assume you have a basic comfort with differential equations, with test particles, with gravitating bodies, with acceleration, and so on. That’s a huge amount of background. And then we leave most of it fixed, and poke hard in a few places, seeing what happens when you change those things around, but leave most of the intellectual edifice unchanged.

This is a good strategy for building insight, if you’re already knowledgeable about a theory. But it’s likely not so helpful for newcomers. Perhaps this is why these questions weren’t discussed in my introductory physics classes! It’d be fun to find an approach that also works for newcomers. The notion of universal force laws is one of the most beautiful and audacious ideas humans have ever developed.

Perhaps one approach could be to write a piece of discovery fiction explaining how the second law could have come to be discovered. I find it truly remarkable just how much Newton and his contemporaries needed to get right. There’s so many different, subtle ideas; each needs to sit in the right relationship to the others. It’s remarkable they were able to bootstrap them all collectively into a useful form.

Finally, let me emphasize that, considered as a theory of physics, Newton’s laws are wrong. At best they’re an approximation to certain parts of a theory we hope may be correct, quantum mechanics. I find it astounding that a theory like quantum mechanics can have inside it another theory, an approximation, also extremely beautiful mathematically, but radically different. It’s like taking Bach, adding some noise, and getting the best of the Beatles out. I wish I understood better why this can happen.

Acknowledgments: Many thanks to David Chapman and Andy Matuschak for the conversation which instigated this essay.