Sunday, 22 October 2017

Applications of higher-order derivatives of position

One of the first things we learn in physics is that velocity is the rate of change of position, acceleration is the rate of change of velocity, and how to figure out the quantities you don't know based on the ones you do. Velocity and acceleration are important throughout physics because of velocity's part in momentum and kinetic energy and acceleration's role in Newton's law of motion. What we don't hear much about are the higher-order derivatives. Here, I'll briefly discuss these quantities and what they're useful for. For more detail than in this post, this paper summarizes a lot of the information and applies it to trampolines and rollercoasters.

Jerk-the third derivative

The rate of change of acceleration with respect to time is called jerk. In highschool, when I was dealing with a lot of acceleration based questions, I imagined that when I got to university I would start seeing jerk-based questions. I was wrong, they never come up. There is a lot of information on the Wikipedia page about jerk, more than I'll get into here.

Most applications of jerk relate to its minimization. Reducing changes in acceleration throughout a trip makes the trip for comfortable, and most engineering for "smoothness" of some sort deals with minimizing jerk. There are a lot of papers on this, many of them having to do with robotics. The challenge is figuring out which joints to move and when in order minimize the jerk of the payload.

There have been some papers published in the American Journal of Physics and related publications about the educational value of studying jerk, and in my paper on minimizing relativistic acceleration, we left jerk minimization as an exercise for the reader.

Snap, Crackle, and Pop (4th, 5th, and 6th derivatives).

The derivatives of jerk are sometimes called, respectively, snap, crackle, and pop. In searching for the origin of these terms (obviously they're taken from the Rice Crispies characters, but the origin of their use in physics), I found a reference in a 1997 paper that stated:
These terms were suggested by J. Codner, E. Francis, T. Bartels, J. Glass, and W. Jefferys, respectively, in response to a question posed on the USENET sci.physics newsgroup.
Doing some sleuthing and getting into weird old internet stuff, I found the current iteration of that newsgroup, which has a lot of John Baez arguing with crackpots, and the thread being referred to may be here. Doesn't quite explain the origin though. They may have just been invented by John Baez.

Whereas jerk tends to be something that is minimized to ensure a smooth trajectory, snap tends to be used for predictive motion. Predicting the motion of wrists for prothetics, of quadcopters for interception, and of cats' eyes when they're watching videos. There was a paper examining the snap of the cosmological scale factor, which may be one of its most fundamental uses.

Crackle is rarely used (based on a literature search), and when it is it again tends to be in predicting human motion. Pop is similar, there is very little written about it (besides "it's called pop!") except for an engineering paper again about predicting motion. Less applied, in the world of physics there is this paper, on integrating the N-body program defines kinematic variables all the way down to pop in its algorithm.

After that, there has been nothing written about the seventh derivative, one paper about using the eighth derivative for satellite orbits, nothing for ninth or tenth, and a 1985 MIT internal memo about AI and eye tracking that mentions the eleventh derivative as an example. I think that is the bottom turtle.

Absement: the Integral of Position

The integral of position is called absement for some reason. If you spend one second standing on a one-meter stool, you will have gained an absement of one meter second. Gas pedals in cars function with absement in mind, your speed depends on how far you depress the pedal and how long you keep it down. There is a musical instrument called a hydraulophone which looks like several adjacent water fountains, and sound is produced when someone pushes down on one of the streams. The tone produced is proportional to how far and how long the stream is depressed. In fact, I would semi-seriously argue that the whole concept of absement just exists to describe hydraulophones and vice versa. There is a big explanation on Wikipedia about how there are higher integrals like "absity and abselleration" and an integral of energy called actergy, but I'm pretty sure someone just made these up on a whim sort of like those weird animal plural names like "a parliament of owls." There are two names, Mann and Janzen, that repeatedly come up when searching these things, so I suspect those are the main promulgators of these words.