Waiter, There's a Derivative in my Cereal

By Sean Carroll | July 1, 2008 8:10 pm

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 Gibbons

Abstract: 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.

  • http://godplaysdice.blogspot.com Isabel Lugo

    For what it’s worth, the “snap, crackle, pop” series isn’t original to that paper; it dates back to at least the late 1990s.

  • Kurt

    According to wikipedia

    A jerk system is a system whose behavior is described by a jerk equation, which is an equation of the form (Sprott, 2003):

    frac{mathrm{d}^3 x}{mathrm{d} t^3}= fleft(frac{mathrm{d}^2 x}{mathrm{d} t^2},frac{mathrm{d} x}{mathrm{d} t},xright).

    I do not think snap, crackle and pop are official names given by physicists.
    What use would the jerk, snap, crackle and pop be in cosmology????

  • http://www.booberfish.com/blog/ GP

    I love the term jerk, and snap, crackle, and pop, are just icing on the cake, even if I’ve only ever seen them used in versions of that footnote. It’s still a good laugh.

  • Tom

    Never used the terms, but have used the equations for non ballistic work, deriving interpolated IMU data from measurements that minimize higher order terms. Been about 15 years since I did the work – so memory is a bit faded.
    However, this was more the engineering side of things than the physics end.

  • http://www.cthisspace.com Claire C Smith

    …interesting cos, always thought it was somehting like, if acceleration is a rate of change of velocity or speed, where speed is a scalar quantity, has size and magnitude – but acceleration is a vector function – has, magnitude and direction – somehow, that the -jerk- was the rate of change of that acceleration, and the -joust- was the constant for the rate of change, of acceleration.

    Sort of utilising all the torque or forces against one another.

    Calculus for fair ground rides.


  • Michael T

    Kinda like a little whimsy in science lest we forget the “quark”?

  • Jason Dick

    Claire, Velocity is a vector (speed plus direction of motion). Position is also a vector (distance from some pre-defined central point plus direction from said point).

  • Jason Dick

    Whoops, forgot to note: acceleration is not necessarily just a change in speed, but can also represent a change in direction. An object moving in a circular path, for instance, may not experience any change in speed, but does experience acceleration to change its direction.

  • Phased Weasel

    I’ve heard of snap, crackle and pop as the derivatives from my physics graduate student friend. He also told me that the units of area of particles, in the context of particle colliders, are “barns”. As in, “Can’t hit the broad side of a barn.” Any truth?

  • joulesm

    oh snap!!

  • http://www.flickr.com/photos/coneslayer/ mph

    Phased Weasel, absolutely true.

  • http://www.theory.caltech.edu/~preskill John Preskill

    How old do you have to be to know the tune that goes with these words?

    “Snap, what a happy sound
    Snap is the happiest sound I found
    You may clap, rap, tap, slap,
    But Snap makes the world go round
    Snap, crackle, pop – Rice Krispies!

    I say it’s Crackle, the crispy sound
    You gotta have Crackle or the clock’s not wound
    Geese cackle, feathers tickle, belts buckle, beets pickle,
    But Crackle makes the world go round
    Snap, crackle, pop – Rice Krispies!

    I insist that Pop’s the sound
    The best is missed unless Pop’s around
    You can’t stop hoppin’ when the cereal’s poppin’
    Pop makes the world go round
    Snap, crackle, pop – Rice Krispies!”

    My friends in college had the words memorized, as I did, so we had some pretty deep bull sessions about whether we preferred Snap, Crackle, or Pop. I’m a Pop man, myself.

  • http://whenindoubtdo.blogspot.com/ Eugene

    leave it to the cosmologists to come up with cute names. I think it has something to do with trying to outdo those darn particle theorists and their damn charms and beauties.

  • Luke

    It would be premature to introduce the 5th derivative, unless you want your paper to end up on the cracklepot pile.

  • brad

    If you want to use higher derivatives, start working on pulsars. There are pulsars for
    which several derivatives of the period have been measured, and these considerations
    are, in fact, necessary to discuss the observations. They probably already have names
    (although snap and pop are actually quite reasonable and I’d be happy to adopt them.
    Crackle less so – jokes are fine, but when they start to confuse the issue, you should stop).

  • KayDubs

    …I think the 5th derivative is called “jounce”. I remember hearing that once somewhere…

  • jim

    Is there a Planck limit (or something like it) to these derivatives? At some order does it become physically meaningless?

  • http://scienceblogs.com/catdynamics Steinn Sigurdsson

    Jounce and Jolt are the next higher derivatives.
    They have been used in pulsar astronomy for many years, and before that in mechanical engineering.

    Read the literature folks!

  • http://blogs.discovermagazine.com/cosmicvariance/sean/ Sean

    “Reading the literature” is so Web 1.0.

  • Aaron

    Position is not a vector, though even in flat space it can be useful to model it as one. Position is an element of an affine space. Adding and scaling positions doesn’t make sense. Subtracting does, and then you want to start talking about torsors — displacement is a vector.

  • Tom Snyder

    Maybe not worth mentioning, but I will anyway. If you’re defining these things as time derivatives of vectors, then a particle in uniform circular motion has velocity, acceleration, jerk, snap, crackle, pop, …(ad inifinitum). (The ratio of the magnitude of any one of these to its predecessor is just the angular velocity of the circular motion.)

    Now, if you’re riding on some ride at six flags and going in uniform circular motion, you will *feel* just a steady inertial force. You will not *feel* any jerk (nor any snap, crackle, or pop). That makes me wonder if jerk, etc. would be better defined in terms of rates of change of magnitudes rather than rates of change of vectors. Not sure (and probably nobody cares).

  • Mike Webster

    Uniform circular motion, it seems, would have a first order time-dependent linear velocity since the direction is constantly changing (I assume by uniform circular motion you mean a constant circular velocity). But the derivative of that would give a constant acceleration and all the higher derivatives of a constant is 0, no?

  • http://www.cthisspace.com Claire C Smith

    Thought it was, speed a scalar, accleration a vector.

    Unless at uniform speed, I would assume that if you accelerate or de-ccelerate on a corner/bend in your car or go in a circle, one of two things, or both will occur,

    a) you are going to get pulled up by the coppers, (unless you say your doing a physics experiment in which they will let you off)

    b) Your car is wanting to go off on another tangent, direction. Usually outward, to change its position constantly due to the force of acceleration. May the force be with you.

    Here I was mainly talking O Level physics stuff. A level goes in to other.

    Wierdly I mentioned the jerk, later I meant the jounce (the other pposter said correctly, Kaydubs it was) I actually said the Joust not Jounce by accident. Using the word wrong I loose a point.

    The initial thing I meant, thought about the analogy of amusement park rides because of the maximumization of forces to create that, “I feel sick effect”.!

    Always been interested in forces, but this is why I like drving too.


  • http://www.cthisspace.com Claire C Smith

    Tom Snyder,

    I thought if the rate of change of speed was acceleration and the rate of change of acceleration was the jerk, the constant for the rate of change of acceleration was the jounce.

    That’s actually what I meant first time.


  • DanO

    There are corresponding silly names for derivatives of momentum as well.

    Momentum -> Force -> Yank -> Tug -> Snatch -> Shake
    Velocity -> Acceleration -> Jerk -> Snap -> Crackle -> Pop

  • Mike Webster

    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 … :) )

  • Tom Snyder

    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.)

  • Lawrence B. Crowell

    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

  • Kaleberg

    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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About Sean Carroll

Sean Carroll is a Senior Research Associate in the Department of Physics at the California Institute of Technology. His research interests include theoretical aspects of cosmology, field theory, and gravitation. His most recent book is The Particle at the End of the Universe, about the Large Hadron Collider and the search for the Higgs boson. Here are some of his favorite blog posts, home page, and email: carroll [at] cosmicvariance.com .


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