One interesting fact that has been missed here is the nature of time reversal. F=ma. Switch on the time reversal machine: t becomes -t. Positions stay the same

One interesting fact that has been missed here is the nature of time reversal. F=ma. Switch on the time reversal machine: t becomes -t. Positions stay the same, but velocities reverse, because dx/d-t -> -dx/dt. BUT accelerations stay the same! dx/d^2t gets a double negative, which is just 1. Gravity attracts in the future and the past. Throw a ball up and it comes back down; reverse the camera and it looks the same. This means that (handwaving) forces are independent of time. And this in turn leads to a time-independent construction of forces - how about as the gradient in space (NOT time) of a potential. This I think is a path to partial insight. A particle moves in a potential, and we can calculate the quantity 'a' from the purely-spatial gradient at the particle's position, without reference to time at all. And we can do this because 'a' is an even derivative of position. This is a crappy argument; at best it argues that it's nicer that f=ma instead of f=mv or f=ma'. Fourth or sixth derivatives are not considered. Still, understanding dynamics as governed by position only is compelling.