From Osher Doctorow
The Riccati Differential equation constraints of Section 110 applied
to y = velocity yield:
1) dy/dt = A(t) + B(t)y + C(t)y^2
and since dy/dt = a (acceleration or a(t), we have with y = v(t)
(velocity):
2) a(t) = A(t) + B(t)v(t) + C(t)v(t)^2
According to Section 110, the early Universe accelerations allowed
would then be either exponential or linear or parabolic (which agrees
with Inflation theory), while the later accelerations can be more
general including elliptic. The Universe seems to grow more flexible
with time roughly speaking, unlike what most people think happens for
human beings.
It might be asked whether we can apply Section 110, which was based on
only forces, to scenarios of mixed force and length/distance and time
variables. It seems to me legitimate to continue in the direction of
more and more generalization of the Riccati Differential equation
which already emerged in Section 110, which in turn allows more
general accelerations, unless evidence to the contrary arises. Once
we know how forces operate from Section 110 in the early Universe in
an exclusively force scenario, then that arguably becomes a model for
later applications of the "Final Equation" (see Section 110 and
earlier). There is nothing that prevents one from restricting
variables to forces only or distances or times only in the later
Universe using the Final Equation.
From another viewpoint, since force classically = ma, it is "natural"
to apply the Riccati Differential equation (1) to acceleration,
leading to (2). We also should use the few hints that we have as to
how forces change with time from the Early Universe, such as
coefficients or variables being replaced by their multiplicative
inverses or else remaining the same, whenever we can unless we obtain
further knowledge.
Osher Doctorow
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