Let's try to look at a common scenario with fresh eyes: I am holding a stone in my hand while I am rotating my arm –for simplicity, let's say around the shoulder. At a certain point, I suddenly let go of the stone and watch it detach from my hand to follow a trajectory in the air, until it falls to the ground somewhere.
From the point of view of classical mechanics, what happens here seems pretty clear. If we care about the future motion of the stone, the only relevant data about the motion of my arm is the position and the velocity of my hand at the moment[1] of letting go. This is, in fact, the definition of letting go: the stone was moving under a constraint up until that moment. That constraint –my clenched hand- made sure that the position and the velocity of the stone matched that of my hand at all times. We can describe the action of the constraint in terms of the application of forces, but it is not clear to me at this stage whether that matters.
Right after letting go, the velocity of the stone obeys this equation:
where Vhand is the velocity of the hand at the moment when the stone was let go, F is the resultant of all forces acting on the stone (none of which are due to the hand), m is the mass of the stone, and t is some small time. If we treat this as a structural causal model with Ft/m as a noise term, we may end up writing Vhand -> Vstone, that is, the velocity of the hand causes that of the stone. Intuitively, this feels right: the hand is driving the stone –at least up until the moment of letting go- and the stone is driven by the hand. Note though that this asymmetric relation does not appear explicitly in our equation: we might as well have written the same equation for the motion of the hand after letting go of the stone, just with a different F and m.
If we consider the next derivative, the acceleration of the stone, we may be led to write along the same lines (note that we are not taking the derivative of the previous equation though, just proceeding by analogy):
where J is a jerk, a term with the dimensions of a derivative of acceleration. For small t, this becomes
But this equation is wrong. The right one –according to classical mechanics and in good agreement with experience- is instead
meaning that the motion of the stone after we let it go is entirely unaffected by the acceleration of the hand at the time we let go. In fact, in this simple model where the act of letting go is instantaneous, the acceleration of the stone as a function of time is discontinuous. This is true for higher order derivatives as well: the stone carries away no information about them.
Why would Nature treat the first derivative of position so differently from the second, the third and the nth? Interestingly, in our naïve pre-Galileian physical intuition, a stone that is let go carries even less information about the motion of the hand that let it go: its free motion must obviously start from the point where it was let go, but there is no requirement that velocity is continuous. Our intuition clearly does not allow a stone to teleport, so its position is understood to be a continuous function of time, but that’s about it. In some naïve depictions the stone reaches rest immediately, and that would make velocity discontinuous.
Classical mechanics fares a little better than our intuition: the stone remembers both the position and the velocity of the hand that let it go, so both end up being continuous functions of time. But what about acceleration? What about jerk, snap, crackle, pop and all the other higher order, unnamed, derivatives of position? As soon as the hand relaxes and the stone is free to go, all that information is lost forever. The motion of a projectile carries only limited information about the motion of the thrower.
It is natural to ask why. Why stop at the first derivative of position? Why not at the second? Or at the thirty-seventh for that matter? Let’s first try a (broken) argument that you have to stop somewhere. Pretend, for the sake of the argument, that the motion of the hand that held the stone is described by an analytic function of time u(t), while the motion of the stone is described by v(t). If we insist that the stone’s function v(t) must match all the derivatives of u(t) up until infinity, then u(t) and v(t) would be the same function: an analytic function equals its series expansion, and the two series would be exactly identical.
This is a general problem with assuming that positions are analytic functions of time: if two such functions coincide over an interval, then they coincide everywhere. In such a world it’s impossible to lose one’s wallet: if my wallet has been moving with me for a finite interval of time, it will move with me forever.
Of course it may simply be the case that assuming analyticity is overkill and physically unjustified. But this seems like a lazy way out. A more promising idea is that the functions describing the positions of two different objects over time may be very similar, but will never be identical: bodies have an extension and never really coincide in space[2]. At any rate, this would force us to better model the process of holding a stone and consequently its opposite, that of letting go. I could for instance imagine that the stone is a point mass bouncing elastically inside a shell.
Back to our original problem: we saw that acceleration is not transmitted. For instance whether the hand is in circular uniform motion around the elbow rather than around the shoulder does not matter for the motion of the stone; what matters is only the linear velocity at the moment of letting the stone go, and this led us to write Vhand -> Vstone in DAG notation.
Now imagine that we were to actually witness an acceleration in the stone after it’s been let go by the hand. No matter what we actually observe –and in particular even if we were to observe that this acceleration matches that of the hand- we would be very unlikely to conclude that Ahand -> Astone. We would rather think that there is an additional external force that acts both on the hand and on the stone, so that for instance Ahand <- G -> Astone. An example of a suitable force would be Earth’s gravity. But once again, why suppose two different causal structures for the first and for the second derivative of position? Why not assume, similarly, that Vhand and Vstone have a common cause?
[1] This moment is well defined: the frame of reference where the stone is at rest is the same as that where the hand is at rest, and the distance between stone and hand is zero. Letting go is an event, corresponding to a single point in space-time.
[2] Note though that the center of mass of my hand may well coincide with that of the stone, because my hand is not convex.
Acceleration has a habit of being discontinuous: https://www.mathpages.com/rr/s4-04/4-04.htm (in this link, starting from rest it jumps to -GM/r^2)
Note that the substack app messes up the equations