# Non-physical Jounce Examples in Nature

What are some good examples of jounce, the fourth derivative of position, in the non-physics arena?

The reason I ask is that A) it's already difficult for a lay to visualize it in the physical arena and B) you never hear of too many examples past the second derivative outside of said physical arena.

Jerk is relatively easy to perceive when one slams on the breaks and then lets them go, and the best way to describe jounce is an amusement park ride since you're always being jerked around.

Outside of physics, it's a little hard to come by, so I'd like to see if there are other ways of perceiving higher order derivatives in other disciplines.

Many thanks in advance!

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In determining stability of nonhyperbolic fixed points of one-dimensional discrete dynamical systems, it may be necessary to evaluate the so-called Schwarzian derivative. The formula for the Schwarzian derivative involves the third derivative of the function in question. –  Gerry Myerson Nov 9 '12 at 5:10
A related question. –  Guess who it is. Nov 9 '12 at 12:00
I added a definition and a link since I'd never heard the word jounce before. I have heard the fourth derivative called whip; Wikipedia says it's also called snap but I was once told that snap is the fifth derivative. I've never heard any of these terms used seriously, more just in the context of "did you know that the fourth derivative has a name"? –  Nate Eldredge Jun 19 '13 at 21:42
Bending an elastic beam helps me understand the fourth derivatives pretty well, for the model uses the fourth derivative w.r.t. space, not time. –  Shuhao Cao Jun 19 '13 at 22:23

## 1 Answer

You write of "examples past the second derivative," so I'm going to assume the third derivative is of interest to you. As I noted in a comment, "In determining stability of nonhyperbolic fixed points of one-dimensional discrete dynamical systems, it may be necessary to evaluate the so-called Schwarzian derivative. The formula for the Schwarzian derivative involves the third derivative of the function in question."

To expand on this a bit:

Given differentiable $f:I\to I$ where $I$ is an interval, and $x$ such that $f(x)=x$, it can be shown that $x$ is an asymptotically stable fixed point of $f$ (roughly: if you start near $x$, and iterate $f$, you converge to $x$) if $|f'(x)|\lt1$, and unstable if $|f'(x)|\gt1$. Things are more complicated if $|f'(x)|=1$ and, in the case $f'(x)=-1$, the determining factor is the value of the Schwarzian derivative evaluated at $x$. You can look up the Schwarzian derivative --- you'll find it involves $f'''(x)$.

EDIT: Maybe you'd prefer Wilson, Matthews, Greasham, Will, and Copeland, Application of fourth derivative absorption spectroscopy to protein quantitation during purification, Anal Biochem. 1989 Oct;182(1):141-5.

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You asked for examples "in the non-physics arena," "outside of said physical arena," "Outside of physics," and I have presented one. If what you really want is examples "where math isn't the perceived prime concern," then you should consider editing the body of your question to say so. –  Gerry Myerson Nov 11 '12 at 23:16

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