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[Phys-L] Re: A problem of motion and derivatives



Bob Sciamanda wrote:

You can concoct for yourself more bizarre (but possible) motions in which
the velocity keeps changing even though (at some time) it and all of its
derivatives but the Nth one are zero, N being as large as you please.

Yes! Bob is correct mathematically and physically; I find Rick's counterarguments
unpersuasive.

If you want to start a collection of "bizarre (but possible) motions", here
is my contribution. I hope it is bizarre enough:
http://www.av8n.com/physics/img48/differentiable.png

That's a graph of the function

-1
f(x) := exp ( --------- ) for all x in the open interval (0,1)
x (1-x)

f(x) := 0 for x=1 and for x=1


And we can define capital F() to be the periodic continuation of f(), i.e.

F(x) := f(x mod 1)

You can easily verify that:
-- F(x) is continuous
-- F(x) is differentiable
-- F(x) is continuously differentiable
-- F(x) is infinitely differentiable.

That is, the Nth derivative of F(x) exists, for any and all values of N.

What's more, *all* of these derivatives are zero at x=0 and indeed whenever
x is an integer.

Hint: Every derivative will be of the form (some exponential) times (some
polynomial), and the exponential will always go to zero and dominate the
polynomial.

If you want to expand F() in a Taylor series around x=0, you're going to be
very disappointed.

===========

It is a useful rule of thumb that physically-interesting quantities can
usually be understood by expanding them to lowest order, i.e. by looking
at a low-order Taylor series approximation. I am very fond of saying
things like
-- To first order, everything is linear.
-- To second order, everything is parabolic.

But it is important to realize that these cute sayings, although "obviously"
true, are not actually strictly true!

The idea that there is "always" a Taylor series is only a rule of thumb.
It is *not* a rigorous result of mathematics. It is not even completely
reliable when restricted to physically-relevant functions. Critical
phenomena provide some spectacularly relevant counterexamples. Over
many decades a lot of smart people (including the likes of Curie and
Landau) got snookered by this. See for example the discussion of figure
36-15 in Feynman volume II. RPF says the observations "fit the theoretical
curve fairly well" but if you actually look at the figure you see a
systematic discrepancy -- a smallish discrepancy, but readily perceptible.

If you had figured out the reasons for this discrepancy, you could have
gotten an all-expense-paid trip to Stockholm.
http://nobelprize.org/physics/laureates/1982/presentation-speech.html
A key step in the solution was realizing that there was no Taylor series
expansion around T=Tc.
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