# How are the higher derivatives (jerk, jounce) of position with respect to time used in physics? [duplicate]

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I don't see them much in any physics lesson or course, but that's probably because I'm not into physics as much. I can understand why velocity and acceleration are useful, but why would someone want to care about, let's say the snap, crackle, or pop derivative of position? They also seem rather complicated. Jerk equations.

## marked as duplicate by JMac, John Rennie, Kyle Kanos, Chris♦May 10 at 16:12

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## 3 Answers

As the equations of motion are of second order, the higher derivatives give no new information (but follow uniquely from the initial conditions of position and velocity), therefore they usually are not discussed. (Note: As Timaeus pointed out there are specific scenarios, e.g. Norton's dome where intial values for the higher order derivates will change the outcome, this has to do with singular solutions of ODE, but these are unstable against infinitesimal perturbations, so I would not consider them physical).

The higher derivatives occur in some engineering applicaitons, usually in the context of safety limitations of something. As the jerk determines the rate of change of accelaration it is relevant when some mechanical device must get into an equilibrium with the apparent force due to acceleration. Intuitively, if the jerk is in a way higher than the resonance frequency nasty things may happen (because instead of finding a new equilibrium the system will instead oscillate and may break).

This gets even more important for controlled systems (such as humans) where the regulation may break on large jerk (as it has no sufficient time to reach a new equilibrium state if the setpoint is changing to fast). See Wikipedia on this.

The jerk does occur in an equation of motion in one case in physics: Radiation dampening of motion. See the Lorentz-Abraham force. This seems to cause problems with causality and energy conservation (as the equation has a solution with infinitely increasing energy if the initial jerk has certain properties). In a physical system these problems are avoided because the initial jerk is not actually free, but bound by the previous motion of the system. (This problem comes about because of the infinte energy density of a point charge, so to say directly from the inconsistency of classical electrodynamics – due to special relativity, which follows from classical electrodynamics, particles must be point-like but classical electrodynamics shows divergencies for point charges).

• Norton's dome has solutions where higher order derivatives can pull out unique solutions. Second order doesn't give you unique solutions for fixed position and velocity unless you add higher order differentiability or continuity or Lipschitz etc. criteria. That these solutions might lie on a subset of measure zero on a phase space could be an argument, but measures on phase spaces aren't a natural given. – Timaeus Jun 11 '15 at 19:06
• Well then feel free to add an answer that discusses that, I will make more clear, that there are very special situations where higher order derivates may influence the outcome. But don't you get a non-deterministic solution for initial jerk of zero in your scenario? (As the singular solution $x = 0$ may branch to the moving solution at any time without violating the equations of motion?). Okay, caring for higher order differentiability matters then (as this will forbid such solutions), but you still do not care for the value of the higher order derivative? – Sebastian Riese Jun 11 '15 at 22:24

Generalising a concept is one of the things people usually look at to get a bigger view of an idea. So one of the natural things to wonder about when you here that if velocity is the time derivative of position and acceleration the time derivative of velocity, then what is the next step? What is the time derivative of acceleration?

So the reason the topic of higher order terms gets mentioned is because it is a natural thing to wonder about and not because they have too much utility (if any) in the physics problems that students are likely to encounter.

There are some issues when the mechanics of human design when it comes to jerk, though.

It seems very likely that a 'big jerk' is one of the things that will make a roller coaster ride unpleasant. I like the feel of acceleration, but not the feel of getting violently thrown around.

Also, back when I was an undergrad, one of the other students in my physics class was trying to figure out a nice way to eliminate the sudden motion of his electric wheel chair when it started from a stop. I hope this problem has been solve by now.

This answer comes from cam synthesis world, as well as roller follower dynamics. Just imagine everything moving is also flexible is some way.

Generally the force applied to a moving part is proportional it it's acceleration and the point of controlling jerk, snap and crackle in a mechanism is so that we smooth out the forces and do not impose vibration harmonics where they can be harmful.

Consider a case where a force applied is instantaneously stopped. You have experience with this if you ridden a bus. If you are holding a drink you can spill it if you don't actively manage it. By doing that you are controlling shape of the forces applied to the cup and hence the derivatives of acceleration.

First level of control is to make acceleration continuous instead of a step function. So now you have constant jerk. But the drink in your cup will still splosh around and to reduce that you need to smooth out the acceleration and ease into the constant jerk time. This is done by now having constant snap and linear jerk.

Finally in carefully engineered cam mechanisms you have areas of linear snap blending two regions which connects two accelerations curves with a cubic spline.

For example see chapter 13 in Lotus Valvetrain Concept Design.