You can learn a lot by reading scientific papers. For example, I’ve known for a while now that the Hubble parameter tells you how fast the universe is expanding — it’s the (conveniently normalized) first derivative of the scale factor, for you calculus-philes out there. The deceleration parameter tells you how fast the universe is decelerating — it’s the second derivative, telling us how the expansion rate is changing with time. (Of course now we know that it’s really accelerating, but we didn’t know that back when the phrases were introduced.)
Less well known, but more amusing, is that the third derivative — how is the acceleration changing as a function of time? — is characterized by the “jerk.” Makes sense, when you think about it — when you jerk at something, you are not just pulling it (causing acceleration), but pulling at it faster and faster. It nevertheless leads to fun if predictable jokes, with this person or that being labeled a “cosmic jerk.”
Do we really need a name for the fourth derivative, telling us how the jerk is changing with time? Apparently we do, as it has been denoted the “snap.” I just learned this from this new paper:
Cosmic Jerk, Snap and Beyond
Authors: Maciej Dunajski, Gary GibbonsAbstract: We clarify the procedure for expressing the Friedmann equation in terms of directly measurable cosmological scalars constructed out of higher derivatives of the scale factor. We carry out this procedure for pure dust, Chaplygin gas and generalised Chaplygin gas energy-momentum tensors. In each case it leads to a constraint on the scalars thus giving rise to a test of General Relativity. We also discuss a formulation of the Friedmann equation as unparametrised geodesic motion and its connection with the Lagrangian treatment of perfect fluids coupled to gravity.
The best part is this footnote:
The analogous expressions involving 5th and 6th derivatives are known as crackle and pop. This terminology goes back to a 1932 advertisement of Kellogg’s Rice Crispies which `merrily snap, crackle and pop in a bowl of milk.’
I suppose there is also some interesting science in there. But now I really want to write a paper that makes use of the 5th time derivative of the scale factor. The words are too delicious to resist.
Hey Claire,
Speed IS a scalar. However, velocity is a vector (where the speed is the magnitude of the vector and the direction is the, umm …, direction of the vector … 🙂 )
Mike Webster:
In uniform circular motion, the acceleration vector has a constant magnitude but a changing direction. As the object moves on the circle, the acceleration vector changes its direction so that it always points toward the center of the circle.
So, the time derivative of the acceleration vector (the jerk) is not zero. For uniform circular motion, jerk is a vector with a constant magnitude and a direction always opposite to the velocity. The rate of change of the jerk gives the snap which is nonzero and points radially outward from the center. Crackle, then, would be parallel to the velocity. Etc.
So, according to DanO, the Force would be centripetal, the Yank would be opposite to the velocity, the Tug would be centrifugal, and on and on and on. (Uuhhh, I’m feeling queasy.)
A periodic system or a rotation is a case of a C^{infty} function, as e^{-ipx} or cos(px) etc have an infinite number of derivatives. In the case where the angular rotation changes there is a derivative of the angular velocity that is nonzero, where this can be C^k.
In a general setting one can have a Lagrangian of the form
L = L(q, q’, q”, …)
for a large number of primes (time derivatives) on q. Lagrangian systems are a case of Finsler geometry, where for L(q, q’) the one form (dq – vdt), for v = q’, defines the horizonal plus vertical portions of the bundle. For a Lagrangian with higher order derivative on q the bundle one-form extends into various independent vector spaces of dimension n. A related issue is the jet bundle which contains a set or sequence of vector spaces corresponding to higher order differentials.
Lawrence B. Crowell
I was taught that the fourth derivative was called inauguration. A friend of mine who worked for an elevator company confirmed this. Elevator people are very sensitive to higher derivatives, a lot like stomachs. The joke I always heard was that inauguration was change of jerk.
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