From what i’ve learned in basic Calculus and Physics there are a handful of levels of differentation and integration.
first integration - area under the graph
Original function - position
first derivative - velocity
second derivative - acceleration
third derivative - jerk
I assume this goes on for infinity in either direction assuming you have enough powers of the variable so what do things like the fifth derivative or the second integrand do.
What’s the question?
Does differentiation and integration extend onto infinity?
Yes as long as you have a variable and not a constant,
What do the levels of differentiation do?
Depends on the circumstances. As I understand it differentation is just rate of change of x where x is any variable. Integration is more complicated though and makes my head hurt sometimes.
I’m not aware of any physical application of derivatives or integrals of higher order. (A second-order anti-derivative would be used if you started with the acceleration and wanted to derive the position function.) One place that higher-order derivatives would be used is in coming up with Taylor series/polynomial respresentations for functions.
After you get off the list in the OP, you don’t see much in the way of physical interpretations.
In basic physics, not much more that this is ever used. In fact, from what I can tell “jerk” mainly exists to answer the question “what’s the third derivative used for”. The basic structure of Newtonian Mechanics pretty much guarantees you’re only ever looking at second order differential equations.
The third derivative, “jerk”, is used a lot to analyze the smoothness of motion in cams, linkages, etc. It’s not good to have jerk going to infinity, which implies bad stresses in the moving object. And no, I don’t recall (from my MS Mech. Eng. days) anything beyond jerk as having a physical meaning.
This site has some good information, as well as a handy poem to remeber upper level derivitieves, as well as force derivitives:
Momentum equals mass times velocity!
Force equals mass times acceleration!
Yank equals mass times jerk!
Tug equals mass times snap!
Snatch equals mass times crackle!
Shake equals mass times pop!!
According to the site the designers of the Hubble had to put limits on the 4th derivitives of motion for various components.
Derivatives of high order are frequently used in the solution of elasticity problems. The governing equation of plane linear elasticity is called the biharmonic equation and involves fourth-order derivatives of displacement. Plate and shell theories go even higher - I recall reading a paper on disk-spindle elastodynamics in which the disk was treated as a plate; the equations included twelfth-order derivatives of displacement.
Okay, I’ll (slightly) recant. Engineering applications use higher order derivatives than second. I was only talking about basic high-school physics, which is pretty much Newtonian Mechanics and E&M for the really smart kids.
We also have our statistical moments:
http://homepages.inf.ed.ac.uk/rbf/CVonline/LOCAL_COPIES/SHUTLER3/node1.html
This has got me wondering if anyone knows offhand an application outside mathematics for fractional-order derivatives. No, I don’t mean things like Sobolev spaces where the functions “have 3 1/2 derivatives”, since really the applications of those aren’t about the 3 1/2th derivative as about the smoothness of its wavelet transforms.
I’ve heard of fractional-order derivatives, but I’ve never seen them defined. I assume that for a sinusoid, where a full derivative is a translation by 1/4 of the wavelength, a fractional derivative would be a translation by a smaller amount. I further assume that for any other function, you first decompose it into sinusoids, translate them, and then re-sum them. Is this correct?
The fact that differentiation acts like a translation on trigonometric functions is accidental, to say the least.
First, pick a suitable space V of functions such that differentiation sends V to itself. C[sup]k/sup won’t work, for instance, because taking a derivative sends a function outside this space in general.
Now, differentiation is a linear transformation in End(V), and higher-order derivatives are powers in this algebra. An “order 1/2” derivative is a square root of the differentiation operator within this algebra. Such transforms are rather unintuitively behaved. For instance, they’re generally nonlocal – the image depends in some sense on “boundary conditions”. I think people have made inroads in thinking of them as acting on so-called “Bornological sheaves”, but IANA Analyst…
There are a bunch of websites that focus on Fractional Calculus applications. I googled “fractional calculus” a few weeks ago when I was interested and a lot came up. Engineering, Physics, and some other types of modeling. It’s over my head, really.
I know the Fractional Integral operator uses the gamma function but my knowledge of higher math is limited (for now).
Sure it uses Gamma. It pretty much has to to construct a square root. What I want to know is what sorts of engineering and physics use it (and need to) rather than what math goes into it.
Why would you stop when you get a constant? That’s when it get trivial.
Not only is jerk (time rate of change of acceleration) used in machine dynamics analysis of high rate-of-change mechanisms as Can Handle the Truth notes, but it also comes up frequently in rocketry design, particularly with multistage booster stacks. During staging events (where one stage is shut down and ejected, and then the next stage fired up) a high jerk at the right pulse rate can instigate a stack vibration resonance which can, in the words of the professionals, totally &#*! things up. For this reason, the timing and pulse parameters during staging are critical. Because it is a third-order derivative, the solutions are fiendishly complicated and not always amenable to explicit solution. This was a big deal with the Saturn V in which the accelerations and jerks within the stack produced what was described as a “pogo-like” motion, different from the compariatively stiffer Titan rockets. Jerk also plays a part in analysis of vehicle (autombile) crash testing and melioration.
I don’t know of any application for fourth order derivatives in mechanical dynamics, but I’m sure someone somewhere has played with the idea, and probably been awarded a Ph.D. for their dissertation on the topic. :dubious:
Stranger
I’m not sure whether you’re talking about classical pogo or a pogo-like structural mode, but pogo is a common bit of jargon so I thought I’d clarify. The Titan III probably had worse classical pogo than the Saturn V. Pogo happens when oscillations in the propellant feed system couple with a structural mode. Since the industry became sensitized to it in the Sixties, it has been an important design consideration in all liquid-fueled launch vehicles. More info here
Huh. Interesting. I hadn’t considered the effect on flowing propellent (virtually all my work has been with multi-stage solids or single-stage hybrids) but that adds another angle in addition to structural vibration. I’ll have do some reading up on the references at the bottom of your link. Thanks.
Stranger