Regarding the “how many derivatives do you go” question, this is related to problems of overfitting — the more terms in hand, the harder it is to get good bounds on their weighting coefficients from a fixed set of data — but I also think the main is a game, very enjoyable. How much behavior can be described by a simpler set of equations than a more powerful?

]]>For more information, see for example:

A. R. Plastino & J. C. Muzzio, “On the use and abuse of Newton’s second law for variable mass problems” (http://articles.adsabs.harvard.edu/full/1992CeMDA..53..227P)

and

J. Peraire & S. Widnal, “Variable Mass Systems: The Rocket Equation” in Dynamics, MIT Course Number 16.07 (http://ocw.mit.edu/courses/aeronautics-and-astronautics/16-07-dynamics-fall-2009/lecture-notes/MIT16_07F09_Lec14.pdf)

]]>Reminds me of “dappled” as in the poem that starts out “Glory be to God for dappled things …”

]]>Though “dampled” has rather a nice flavor, sorry to encourage removing it.

According to Google it’s an occasional typo for “sample” (probably due to the proximity of the “d” and “s” keys), as well as for “damped” (more speculatively due to a nearby mental groove for “sampled”?).

More on topic, I look forward to more on the math of vibrations. As you point out, being exposed to it and really getting it are often two entirely different events.

The parallelism between mass, spring, and damping and inductance, capacitance, and resistance is indeed elegant. There was quite an industry in analog computers to model mechanical systems, once upon a time. ]]>

Vijay: There can be higher order effects, a spring stretched too far or a dash pot subjected to a very high (or low) velocity, but (at least for engineering) we generally try to stay in the region where the 2nd order equations are a good representation. Things get messy quickly when you stray away.

If one were modeling a proto- solar system the changing mass of the planets would give you u”’ to deal with, as might loss of water ice for a comet.

]]>If I throw a stone in the air, I can see it physically describe the parabolic arc and that makes the underlying math extremely tangible for me. I guess I’m looking for a similarly tangible explanation of why these systems are second order differentials.

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