Practical Applications for the 5th and 6th Derivatives?

[Sanguine]

Recently in my Maths class, it came up that there was not much of a practical application regarding the 3rd and 4th derivatives. After some discussion, we discovered that there was in fact a practical application, the 'Jerk' and 'Jounce' of a Distance/Time graph. In KSP terms, this can relate to a rocket - the 1st is Velocity, 2nd is Acceleration, 3rd is change of acceleration due to consumption of fuel, and the 4th is the change of acceleration due to ISP. At least, that makes sense.

However, with the 5th and 6th derivatives, it is harder to picture. The only thing I can think of is perhaps air resistance, but after MaxQ this isn't much of an issue anyway. So, I guess the question is, are there any practical applications for the 5th and 6th derivatives regarding physics?

[K^2]

Not quite as direclty. The reason is that if your dynamics includes 3rd/4th order derivatives, you grind your teeth, increase step resolution, and get results. But higher order derivatives in differential equations make any numerical method cry. So even if you come up with a model that has fifth and higher derivatives, it will probably be useless for anything practical.

But they do come up. Perturbation theory is probably the cleanest example. Both quantum and classical. And if we look at it in a more abstract way, just as whether there is use of higher derivatives, then sure. I've had to work with functions that involved sums of infinitely many orders of derivatives. (Which, thankfully, simplified.) And I've even came across some applications for fractional derivatives while trying to extend certain integer methods solutions to real numbers.

[PB666]

Recently in my Maths class, it came up that there was not much of a practical application regarding the 3rd and 4th derivatives. After some discussion, we discovered that there was in fact a practical application, the 'Jerk' and 'Jounce' of a Distance/Time graph. In KSP terms, this can relate to a rocket - the 1st is Velocity, 2nd is Acceleration, 3rd is change of acceleration due to consumption of fuel, and the 4th is the change of acceleration due to ISP. At least, that makes sense.

However, with the 5th and 6th derivatives, it is harder to picture. The only thing I can think of is perhaps air resistance, but after MaxQ this isn't much of an issue anyway. So, I guess the question is, are there any practical applications for the 5th and 6th derivatives regarding physics?

Cant say directly to derivatives but in statistics the first order is you average, second order is your variance, third order is your assymetry and forth order is kurtosis.

The last two unfortunately are not applied as often as they should be. Many random probability assessments begin with the assumption of normality, and these two can be used to check for non-normality in the distribution. There are often simple ways of correcting non-normality, if the data is consistently higher than 1, then a log transformation frequently takes care of positve skewing and much of the kurtosis at once.

Positive skewing is often observed in biological and natural responses.

How this plays out technically is that if two groups have different means and the data is skewed positively, then the group with the smaller mean will have less variance, and the higher group will have higer variance. The students T test is so

-so suited for comparison with unequal variance, and so if substntial difference in variance is detected you should then opt to use Welches approximation test. Using the approximation when there is no difference in variance can result in artifactually low random p-values, particularly if one group is much smaller than the second.

[Yasmy]

It is true that third and forth time derivatives don't normally have much of a practical application. This is largely because mechanics is usually completely determined by first and second time derivatives. You may wish to look at the jerk and jounce to characterize your system, but you don't need them to solve the equations of motion. You don't need jerk and jounce in KSP to simulate your rocket. Unless the higher order derivatives enter the differential equations describing your system, you generally have no need for them.

Like K^2 said, in perturbation theory, you can have an infinite series of higher order derivatives. These may be time derivatives, spacial derivatives or derivatives in some abstract quantity. Sometimes you can find a sum for the infinite series without taking derivatives. More commonly, the higher order terms can be ignored because you can prove that successively higher order derivative terms contribute a negligible amount compared to lower order derivatives.

If you can simplify your system by throwing out higher order terms, and still get the same answer to as high of precision as you need, there is no practical use for the higher order terms.

I often work with elastic deformation models of cell membranes with second, third and forth order spacial derivatives. The derivatives are needed to characterize the energies associated with bending, compression, stretching and curvature of an elastic layer. You could use the same model to describe the energy associated with bending or deforming a mattress.

The reason why there are no fifth and sixth order terms in many elastic models is that for normal deformations, the shape and energy of the deformed object are very well approximated by the forth order model. By the time you deform your system enough that the simple forth order model breaks down, you often also start breaking your physical system. (I.e., "plastic" deformations which permanently change the shape of your object by kinking or breaking in pieces.)

Some viscoelastic fluid models use 5th order derivatives.

The rule in physics is simple: Use the simplest model or theory which matches experiment. If you have no need for higher derivatives in your theory, don't use them. If you do, do.

[arkie87]

3rd is change of acceleration due to consumption of fuel, and the 4th is the change of acceleration due to ISP. At least, that makes sense.

Why is fourth acceleration due to change in ISP?

If it is a change in acceleration, it is still "jerk", no? It would have to be a change in jerk to be a jounce.

Edited August 20, 2015 by arkie87

[Heimdall5008]

There are some effects in advanced, modern or quantum optics that require high order derivatives to explain.

These effects are normally so small, that the will be neglected, but under some really special setups you can make these effecte measurable