If you can take an infinite number of derivatives of position, then why is it so hard to visualize/think of real world examples beyond the jerk? Does the function essentially become meaningless to the physical world?

Question

If you can take an infinite number of derivatives of position, then why is it so hard to visualize/think of real world examples beyond the jerk? Does the function essentially become meaningless to the physical world?

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Answer 1

Each derivative is a correlate of something real. The zeroth derivative is position itself. The first derivative is velocity, the second acceleration.

Position is the distance between things, constant motion is what happens when you’re interacting with no one, acceleration happens when there’s energy being converted at a constant rate, jerk happens when you’ve just come into contact with something else that’s accelerating, or accelerating at an accelerating pace (like say a muscle is contracting but its contraction is getting stronger as it approaches the midpoint of the contraction).

To find physical analogies of higher order derivatives, we need more and more elaborate physical setups.

And the physical mechanisms we have around us just max out at a certain complexity, because there’s not much to be gained from having evolved higher complexity.

Animals for instance have bones with muscles attached and the system only needs so much complexity to get around the environment which is all mostly static a pervaded with a gravitational field that provides first order acceleration. We have our muscles which can accelerate our bones at a variable rate (more effort, more adrenaline, etc allows us to contract our muscles quicker). That gives us a one-up on gravity.

I guess a mechanism like a whip might introduce higher order derivatives, ie accelerating at an accelerating rate, because you’ve got a wave traveling to the end of a line and that’s sort of like a singularity right there. As the wavelength approaches zero the frequency approaches infinity causing the tip of the whip to move faster than any muscle could move it.

There must be animals that use that kind of action in their bodies. I would bet money that wherever they do it’s for the purposes of fighting, because fighting introduces a new evolutionary bar that continuously raises as the animals you’re fighting against evolve.

So in a way, evolution itself is the source of higher-order derivatives, because evolution is filtered by systems containing the lower-order derivatives, and evolution rewards the edge.