Re: Dynamical Systems and Expansion-Contraction

>>From Osher Doctorow

The homothetic vector equation of Coley (1999):

1) L_z(guv) = 2guv

with L_z the Lie derivative along z, which I mentioned earlier as
formally a tensor Riccati "Differential" equation, is obviously
analogous to the special case of the Riccati Differential equation
which is exponential Expansion-Contraction:

2) dy/dt = ky (k > 0 is unrestricted expansion, k < 0 unrestricted
contraction)

But what do the Lie derivative have to do with Expansion-Contraction by
itself?

The answer is provided very accessibly on the internet from a rather
surprising place, the University of Houston, in particular Professor
Emeritus R. M. Kiehn, a topologist who points out in the paper "Why
topology?" (accessible under those keywords or at
http://www22.pair.com/csdc/ed3/ed3fre2.htm or under his "Applications
of exterior differential forms" A University of Houston Physics course
on the Internet Physics 7397,
http://www22.pair.com/csdc/ed3/ed3homep.htm) contrasts the Covariant
Derivative and the Lie Derivative essentially as follows:

3) The Covariant derivative changes size and bends but doesn't stretch,
preserving tensors
4) The Lie derivative adds stretching and so changing size and shape to
the Covariant deriv.

In the absence of both (3) and (4), uniform expansion changes size but
not shape of objects.

Neither (3) nor (4) changes the number of holes, and that is where
Kiene's topology comes in. By the way, his paper originally comes from
the 1980s but was updated in the 2000s.

Osher Doctorow

.

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