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Re: [Phys-l] solving an energy equation



Carl Mungan wrote:

This appears to me to be some sort of defect in the way Mathematica is coded. Or am I missing something?

Mathematica, like most computer programs, is something of an idiot
savant. Mostly it does what it is told. It is not good at "thinking",
and certianly not good at "thinking outside the box".

The features of Mathematica are like a box full of Legos. You can put
the parts together to do various things. You have the freedom to build
things that work ... and to build things that don't work.

f[t] := (x'[t])^2 == 1 - 2x[t]
g[t] := D[f[t], t]

I'm not supplying any new information

But you did supply new information! You expressed the fact that x' is
differentiable. This is totally nontrivial. It may be obvious to *you*
based on your well-trained intuition that x' is differentiable, but it
is not obvious to the idiot savant. Mathematically speaking, there are
innumerable solutions to the "f" equation that involve suddenly changing
the sign of x' at arbitrary places.

This is related to my earlier remarks about first-order versus second-order
Markoff processes. Often something that is second-order Markovian in one
set of variables becomes first-order in a larger set of variables. The
"g" equation introduces a new variable, namely x'', that was not present
in the "f" equation. This may seem like a small thing, but it provokes
a profound change in the way Mathematica handles the equation.