What do Newton's laws really say?

I recently read this Feynman quote:

Study hard what interests you the most in the most undisciplined, irreverent and original manner possible.

And decided to express some long-suppressed irreverance on this blog: Newton’s laws of motion are a piece of terrible writing. They do not really encapsulate the work that Newton had actually done. If I was a coauthor of the paper with him, I would write a comment in red in the overleaf. If this was a github pull request, I would refuse to merge it until the laws were rewritten. For reference, here are the laws from Wikipedia:

1. A body remains at rest, or in motion at a constant speed in a straight line, except insofar as it is acted upon by a force.
2. At any instant of time, the net force on a body is equal to the body’s acceleration multiplied by its mass or, equivalently, the rate at which the body’s momentum is changing with time.
3. If two bodies exert forces on each other, these forces have the same magnitude but opposite directions.

Let’s first start with the obvious issue here. The first law sounds like a weaker form of the second law? The second law already says \(F = ma\). If you assume \(F=0\), then as long as the body exists (and thus has non-zero mass), the only possibility is that \(a = 0\). And by the definition of acceleration, \(a = 0\) implies that the velocity does not change. So law #1 is actually implied by law #2. This is like if fight club actually had two rules. First rule: do not talk about fight club. Second rule: actually, all talking is prohibited. I can understand wanting to draw attention to a specific consequence of a law you formulated. For example, just saying \(F=ma\) may not always immediately convey to the reader all of its consequences, and you may want to draw their attention to how the absence of force means velocity does not change. But in today’s parlance, such a statement would be considered a corollary and I wouldn’t mind if law #1 was actually stated as a corollary.

The second issue is more subtle but also bigger. To explain, I need to first take a step back. The thinking in Newton’s time was that motion was the most fundamental thing to explain and the whole universe was simply a large number of tiny things moving around. Sometimes the tiny things got together and formed big things and other times they remained tiny, but the belief was that any phenomenon could be explained by the exact motion of the big and small things constituting it. This program did appear to be successful for a long time. The movements of celestial objects were explained, a fairly reasonable theory of electricity was formulated that relied on charged particles moving under electrostatic forces, and even phenomena like heat and cold were explained using the motion of tiny molecules bumping into each other.

So let’s try to put ourselves in Newton’s shoes and try to formulate a theory of motion, with the hope that once we have a theory of motion, everything will be explainable. Our task is this: can we formulate a theory that can predict the trajectories of objects that are moving around? You see some objects in space, and you are allowed to measure any properties of the objects and the space they are in, and with this information you should be able to predict where the objects will be at each point in time in the future. Newton’s claim was that all you needed to do was to measure the object’s mass and the force acting on it and that would tell you its acceleration and you could integrate that twice to get its position. And now I can state my second issue with the laws: up until now, force and mass are both made up quantities and thus the first two laws have basically no content! Or in other words, if the first two laws were all you were equipped with, and you saw a new kind of trajectory being followed by a collection of objects, you could simiply “explain” it by fitting whatever \(F\) and \(m\) resulted in that particular trajectory. So, for example, if you carefully observe an object for an hour and notice that its acceleration follows \(a = x^2 + y^2 - \sqrt{t}\), you say “ah, this is because the object has unit mass and it’s being acted on by a force \(F = x^2 + y^2 - \sqrt{t}\). In other words, there are no trajectories that are ruled out by these laws. It doesn’t exactly qualify to be a law if it allows every possible behaviour, does it?

I am being a little facetious here. Force and mass weren’t completely made up quantitites. They were defined based on intuitions from the everyday world. We do observe that to move an object we have to go and push (or pull) it. Force is literally the word used to describe this act. We need to apply force to move things. And so Newton thought that if something was accelerating, something must have been applying a force on it. Mass is our intuitive understanding of how difficult it is to move something. Heavier objects are harder to move than lighter objects. And so it was hoped that mass represented a similar quantitity in Newton’s laws. While these intuitions do not give us a precise way of measuring mass or force directly, they do help us make the necessary assumptions that give Newton’s laws some actual predictive power. For example, based on our intuition about mass, we would probably postulate that the mass of an object remained constant with time. Similarly, based on our intuition about force, we would probably have opinions on the variables that a force is a function of. Suppose we want to predict the motion of an object tied to one end of a spring whose other end is tied to a wall. We might think that the force applied by the spring on the object is a function of how stretched the spring is and nothing else. Thus if we tie the object to the spring, stretch out the spring, and let it go, we would expect that the spring would start its oscillatory motion and that every time it reached a stretch of \(x\), the acceleration of the object would be \(f(x)\) for some fixed function \(f\). This is precisely because we think that: a) the mass of the object doesn’t change, and b) the spring force is a function only of the stretch \(x\). Thus with these two assumptions, Newton’s first two laws do rule out several trajectories. In particular, they say that we cannot see a trajectory where the spring attains the same stretch \(x\) twice but the object has a different acceleration each time. This can then be used to come up with predictions. We can measure several \((x, a)\) pairs, where \(x\) is the stretch and \(a\) is the acceleration of the object, and we can fit a curve through the points we observe. When we do that we will get something like \(a = -kx\) for some constant \(k\). And this equation lets us make predictions about what the acceleration will be of this specific object when tied to this specific spring and left at a stretch of \(x\).

By the way, one of Newton’s many contributions was to use his scheme to predict motion of celestial objects. In particular, you can postulate that planets move because the sun is exerting a force on them which depends on: a) distance \(r\) of the planet from the sun, and b) some other properties of the planet that stay constant with time. Now you can point your telescope at a specific planet, say Saturn, and record many \((r, a)\) pairs where \(a\) is Saturn’s acceleration when it’s at distance \(r\). When you fit a curve through the data, you will find that \(a = C\cdot\frac{1}{r^2}\), for some constant \(C\) that depends on those other properties of Saturn that the force is supposed to depend on. This observation leads to the famous inverse-squared law of gravity. However, the weird thing is that if you pick another planet, say Jupiter and run the same experiment, you end up with the exact same expression \(a = C\cdot\frac{1}{r^2}\) (with the same value for \(C\)). You can run this with any number of planets, or even asteroids going around the sun and end up with the exact same expression for all of them. But \(C\) was supposed to depend on properties of the planet! Does this mean all planets are identical when it comes to gravity? In fact, the coincidence that makes this happen is that the property of the planet that this force is supposed to depend on is exactly the planet’s mass. In fact, you can write \(F = C'\cdot\frac{m}{r^2}\) where \(m\) is the mass of the planet. Since \(a = F/m\), the \(m\) cancels out and you end up with an expression that does not depend on \(m\). The thing that a force depends on, and the thing that determines acceleration given a force are the same thing? This is too much of a coincidence! Honestly I feel that this should have set off alarm bells in the apple orchards of Cambridge and the “force” theory of motion should have sounded suspicious. But I guess Newton had done more than enough for one life.

So far we have seen that even though force and mass are made up quantities, if we assume some additional structure on them that were not specified by Newton’s laws, the laws do start to characterize the kinds of motion that are legal in the universe. However, the third law changes that by removing the need to have a lot of extra structure. We still need to assume that mass doesn’t change (and perhaps we can chalk that up to common sense) but we don’t need to assume anything extra about the force. The third law provides that extra constraint that needs to be satisfied by all forces. It basically states that force isn’t a property of one object but of two objects. It’s not that one object exerts a force on another object but that two objects exert a force on each other. It’s not that the sun pulls Saturn towards itself, but that the sun and Saturn pull each other towards themselves. Thus given any two objects \(A\) and \(B\), you get one number (not two) \(F\) describing the force between them. And the nature of \(F\) is that the direction in which it acts on \(A\) is the opposite of the direction in which it acts on \(B\). So now imagine two objects that are in motion due to the force between them. Suppose that there are no other forces acting on them currently. Then if we combine the second and third laws, we get that \(m_1a_1 = -m_2a_2\) at all time, where \(m_1, m_2\) are the masses and \(a_1, a_2\) are the accelerations of the two objects. This can be rearranged to say \(m_1a_1 + m_2a_2 = 0\). Realizing that acceleration is the derivative of velocity wrt time, we get \(\frac{d}{dt}(m_1v_1 + m_2v_2)=0\), which gives us the familiar law of conservation of momentum. That is, the total momentum \(m_1v_1 + m_2v_2\) of the system does not change with time. This is a pretty strong requirement on what trajectories are considered legal by nature’s laws.

The characterization in terms of momentum still refers to mass, which is a made up quantity. So what exact trajectories does it rule out? Can we still explain any trajectories by simply fitting appropriate values of mass in the equations? Imagine two objects \(A\) and \(B\) moving under the influence of each other. Suppose there are no other external forces acting on them. You can then observe their velocities \(v_1(t)\) and \(v_2(t)\) for several values of \(t\). Suppose you measure it for \(k\) different values. Conservation of momentum gives you \(k-1\) equations, but we have only two unknowns: \(m_1\) and \(m_2\). Newton’s laws say that these large number of equations in two variables are consistent with each other.

A law that I think is missing from Newton’s original list is that forces are additive. So if \(A\) and \(B\) exert forces \(F\) and \(-F\) on each other, and \(A\) and \(C\) exert forces \(F'\) and \(-F'\) on each other, then the total force on \(A\) will be \(F+F'\). This isn’t obvious and one could imagine complex non-linear interactions taking place, but it is indeed the case that complex non-linear interactions do not occur and forces add up nicely.

So I would probably write the laws of motion like this:

1. The universe is made up of a large number of objects moving around under each other’s influence.
2. Each object \(i\) has associated with it a quantity \(m_i\) called mass that does not change with time.
3. Each pair \((i, j)\) of objects has associated with it a quantity \(F_{ij}\) called force that can change with time but must always satisfy the property that \(F_{ij} = -F_{ji}\).
4. At any point in time, the acceleration of object \(i\) can be calculated as \(a_i = F_i/m_i\) where \(F_i = \sum_j F_{ij}\).