How would mechanics be different if $F=mx'''$ instead of $F=ma$? I feel like I have intuition about what kind of place an $F=mv$ world would be like, since, after all, this is part of our everyday experience dominated by friction. Let me make my question more specific.
If I pick a ball attached to the end of a string in a $F=mv$ world, I cannot make it move in uniform circular motion by swinging it around. I can separate the ball and the end of string by pulling the two apart, but then pulling on the string simply pulls the ball towards me. $F=mv$ is like living in a very viscous liquid. I've heard that at a very small scale (like that of bacteria), water is like this and this has consequences for the propulsion of microorganisms.
In reality, it is possible to give an initial velocity to the ball and then to tug on the string to rotate the velocity vector so that it is moving in uniform circular motion.
Please correct me if any of the above is wrong.
But what happens if $F=mx'''$? I have no intuition about what would happen with the string and ball example in this case. You could give the ball a constant velocity and zero initial acceleration. But then how does it move constrained by a string that is applying a force which changes acceleration. Would it spiral? What kind of setup of forces could force a particle into circular motion? Is there any way to gain intuition about the consequences of $F=mx'''$?