More on Newtonian velocity

Here is a big picture story about Newtonian mechanics: The state of the system at all times t > t₀ is explained by the initial conditions of the system at t₀ and the prevalent forces.

But what are the initial conditions? They include position and velocity. But now here is a problem. The standard definition of velocity is that it is the time-derivative of position. But the time-derivative of position at t₀ logically depends not just on the position at t₀ but also on the position at nearby times earlier and later than t₀. That means that the evolution of the system at times t > t₀ is explained by data that includes information on the state of the system at times later than t₀. This seems explanatorily circular and unacceptable.

There is an easy mathematical fix for this. Instead of defining the velocity as the time-derivative position, we define the velocity as the left time-derivative of position: v(t)=lim_{h → 0−}(x(t + h)−x(t))/h. Now the initial conditions at t₀ logically depend only on what happens at t₀ and at earlier times.

This fixed Newtonian story still has a serious problem. Suppose that the system is created at time t₀ so there are no earlier times. The time-derivative at t₀ is then undefined, there is no velocity at t₀, and Newtonian evolution cannot be explained any more.

Here’s another, more abstruse, problem with the fixed Newtonian story. Suppose I am in a region of space with no forces, and I have been sitting for an hour preceding noon in the same place. Then at noon God teleports me two meters to the right along the x-axis, so that at all times before noon my position is x₀ and at noon it is x₀ + 2. Suppose, further, that the teleportation is the only miracle God does. God doesn’t change any other properties of me besides position, and God lets nature take over at all times after noon.

What will happen to me after noon? Well, on the fixed Newtonian story, my velocity at noon is the left-derivative of position, i.e., lim_{h → 0−}(2 − 0)/(0 − h)= + ∞. Since there are no prevailing forces, my acceleration is zero, and so my velocity stays unchanged. Hence, at all times after noon, I have infinite velocity along the x-axis, and so at all times after noon I end up at distance infinity from where I was—which seems to make no sense at all!

So the left-derivative fix of the Newtonian story doesn’t seem right, either, at least in this miracle case.

My preference to both the original Newtonian story and the fixed story is to take velocity (or perhaps momentum) to be a fundamental physical quantity that is not defined as the derivative, or even left derivative, of position.

The rest is technicalities. Maybe we could now take Newton’s Second Law to be:

1. ∂_t⁺v(t)=F/m,

where ∂_t⁺ is the right (!) time-derivative, and add two new laws of nature:

2. ∂_t⁺x(t)=v(t), and

3. x(t) and v(t) are both left (!) continuous.

Now, (2) is an explicit law of nature about the interaction of velocity and position rather than a definition of velocity. On this picture, here’s what happens in the teleportation case. Before noon, my velocity is zero and my position is x₀. Because I supposed that the only thing that God miraculously affects is my position, my velocity is still zero at noon, even though my position is now x₀ + 2. And I think (by the answer to this), laws (1), (2) and (3) ensure that if there are no further miracles, I remain at x₀ + 2 in the absence of external forces. The miraculous teleportation violates (2) and (3) at noon and at no other times.

But of course this is all on the false premise of Newtonian mechanics.