Science > Physics > Rotation vs Expansion-Contraction 12: More Raychaudhuri
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Science > Physics |
| User: |
"OsherD" |
| Date: |
26 Jun 2005 09:33:03 AM |
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Rotation vs Expansion-Contraction 12: More Raychaudhuri |
From Osher Doctorow
The shear scalar in the Raychaudhuri Equation brings us in contact with
one of the most remarkable topics in differential equations, namely
autonomous systems of differential equations.
The "simple" but "wrong" answer to incorporating the shear scalar into
the Riccati Differential Equation with possibly varying time
coefficients is the answer using generalized Hooke's law relating
stress and strain which for example for scalar y and constant mass has
the form y" = ky which is then equivalent to a system of first order
equations where y" is the second time derivative of y. The answer is
"wrong" because, like algebra in general, it includes too many things -
it would make linear differential equations essentially equivalent to
equation systems of Riccati Differential Equation type or similar to
those, which although it would be an interesting "explanation" of QM
linearity, would nevertheless have the difficulty of including
curvilinear motion in the same class as Expansion-Contraction and
rotation.
Yet the equation:
1) y" = ky (prime denotes derivative with respect to time)
is in fact "distinguished" among second order differential equations by
being the simplest generalization of the Riccati Differential Equation
(with coefficient 0 of the y^2 term) to higher derivatives. Provided
that we not allow further generalizations of this type (which would
result in making linear differential equations equivalent to Riccati
Differential Equation systems ultimately), let's make this definition.
Definition. An R2-Generalized (R2G for short) Riccati Differential
Equation is one or both of the equations:
2) dy/dt = A(t) + B(t)y + C(t)y^2
3) Dtt(y) = ky (where Dtt is the second time derivative of y)
Of course, via Hooke's law, we can also regard Superstring/Brane Theory
as now contained in Riccati Diferential Equation theory and hence PI.
Osher Doctorow
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| User: "OsherD" |
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| Title: Re: Rotation vs Expansion-Contraction 12: More Raychaudhuri |
26 Jun 2005 09:44:20 AM |
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From Osher Doctorow
I should say that Superstring/Brane theory is contained in R2G-Riccati
Differential Equation theory. If any reader wants to see the details
of this beyond scalars, I recommend it for him/her as an excellent
exercise for the 4th of July :>)
Osher Doctorow
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| User: "OsherD" |
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| Title: Re: Rotation vs Expansion-Contraction 12: More Raychaudhuri |
26 Jun 2005 09:54:44 AM |
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From Osher Doctorow
The reader who wants to actually do the proof as an exercise should
probably start with the constant mass situation as I did. Hint :>) You
could next try evaluating F = kx with F = d(mv)/dt and m variable.
Osher Doctorow
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