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From: OsherD on 19 Mar 2010 00:14
From Osher Doctorow

In view of the last few posts, let xF be the conserved product of
linear expansion-contraction force's magnitude F and displacement x
along the axis of F. For conservation of energy we have:

1) Dt(xF) = 0 = Dt(x Dtt(x)) if F = ma = mDtt(x) for m constant or
approximately constant.

This yields:

2) Dt(x Dtt(x)) = x Dttt(x) + Dt(x) Dtt(x) = 0

or equivalently:

3) Dt(x) Dtt(x) = - x Dttt(x)

which consists of two oppositely directed ("dual") Riccati
Differential Equations without the x^2 term:

4) Dt(x) = -xA1(t), A1(t) = Dttt(x)/Dt(x)
5) Dttt(x) = - Dtt(x) A2(t), A2(t) = Dt(x)/x

Notice how simple (4) and (5) are for A1(t) and A2(t) constants or
even simple monomial functions of time t. We can even find A1(t) and
A2(t) such that an x^2 factor for example replaces the x factor in the
right hand side of (4) with the other factor constant, etc.

A pair of dual Riccati Equations such as the above are somewhat
similar to have a pair of linked Logistic Differential Equations in x
and t in the sense of "feedback".

Osher Doctorow


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