Are there any physical models that compute higher-order derivatives of position than jerk, snap, crackle, pop, etc.? Can we measure higher-order, non-zero components?
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1$\begingroup$ What do you understand on "physical observability"? If I log my trajectory with a GPS logger mobile app, can we consider it as physically observed? $\endgroup$– peterhApr 30, 2017 at 23:19
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2$\begingroup$ For a periodic motion, you can compute an infinite number of derivatives. Depending on what you mean by "measure," you can also measure them. For example measure position, or one of its deriviatives, in the time domain and do a Fourier transform to calculate the others. Whether there is any practical use for the higher derivatives is a different question. $\endgroup$– alephzeroApr 30, 2017 at 23:43
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They are harder to measure and estimate. Keep in mind, that a higher order derivatives are higher order nonlinearities in the system model. You can see that since Taylor series expansion has higher order derivatives for higher order nonlinearities in the series.
For instance , in estimating the parameters of a trajectory you measure position at various points. For higher order nonlinearities you have to measure more and more points. The statistical errors also contribute more for the higher orders. This has been developed well in nonlinear filtering and estimation processes.
See the wiki article on nonlinear (extended) Kalman filters at https://en.wikipedia.org/wiki/Extended_Kalman_filter
There are other techniques, but bottom line the higher the nonlinearities, the harder to estimate -- meaning the higher order derivatives. Same happens when you model any physics as expansions in some parameter.
A typical paper explaining those things is at http://adsabs.harvard.edu/abs/2014PhDT........67G
Probably you can Google a lot on this, and find more in dsp stack exchange. But even in physics, the detection of weak effects like gravitational waves models and estimation need to be used. And you always want to model around the point where you have the strongest effect.
