Science > Physics > Independent/Dependent Phases 23: Generalized Dimensional Analysis With 1/2
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Science > Physics |
| User: |
"OsherD" |
| Date: |
15 Oct 2005 08:54:29 AM |
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Independent/Dependent Phases 23: Generalized Dimensional Analysis With 1/2 |
From Osher Doctorow
Mechanical variables almost all have dimensional exponents 0, 1, 2, or
3 aside from dimensional constants. This gives us an interesting clue
as to how to get 1/2 (or 1/3, etc.) from them: take partial derivatives
to get respectively 2 and 1 (or 3 and 1, etc.) and then divide
variables.
As an example, curvature C has dimension L^(-1) where L is the
dimension of length. Energy E has dimensions ML^2 T^(-2) where M is
the dimension of mass, T is the dimension of time. So we get with DL
the partial derivative with respect to L:
1) DL(C) = -L^(-2)
2) DL(E) = 2MLT^(-2)
3) DL(C)/DL(E) = -(1/2)L^(-2)M^(-1)L^(-1)T^2
In (3), I don't bother to reduce or simplify the right hand side since
the point is that (1/2) is the numerical factor up to +/-.
It turns out that momentum, with dimensions MLT^(-1), is one of the
most versatile numerators in this method either with DL or DT (the
partial derivative with regard to time), the denominators being
typically energy (E), power (P), angular momentum, kinematic viscosity,
force when DL is used, or force, pressure, enery, surface tension,
modulus of elasticity, compressibility for example when DT is used.
Compressibility and curvature act as numerators in the same scenarios
as momentum for DL, while for DT angular momentum and dynamic or
kinematic viscosity act as numerators.
There isn't a big problem in regarding 1/2 as a phase boundary because
we can always propose inequalities with ratios, with or without (new)
dimensional constants.
Osher Doctorow
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| User: "OsherD" |
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| Title: Re: Independent/Dependent Phases 23: Generalized Dimensional Analysis With 1/2 |
15 Oct 2005 09:08:28 AM |
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From Osher Doctorow
Notice the remarkable fact that curvature, viscosity, and surface
tension are extremely important in some of the main unsolved problems
in physics and/or astrophysics and engineering including the
Navier-Stokes equations, quantum gravity, etc. Momentum is of course
useful almost everywhere.
As for the motivation for calculating DL or DT aside from getting the
factor 1/2, if we regard dimensions of a variable as reflective or
indicative of its (probable) causation, then by Birkhoff causation as
embodied in differential equations via the derivative (see my previous
postings) and the fact that the Riccati Differential equation of
expansion-contraction involves the time derivative (see also my
previous postings) and is the fundamental equation of
expansion-contraction, we can regard variables as implicitly involving
time derivatives and so taking DL of a variable is actually a DLDT or
DL(DT) composition and so a mixed partial derivative (see my previous
postings). We can generalize this idea in doing the same with DT
instead of DL at the beginning.
Osher Doctorow
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| User: "OsherD" |
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| Title: Re: Independent/Dependent Phases 23: Generalized Dimensional Analysis With 1/2 |
15 Oct 2005 09:34:42 AM |
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From Osher Doctorow
What do I mean by it being easy to get inequalities of variables rather
than just equalities involving 1/2? Well, if the ratios of the DT or
DL of two variables involves the factor 1/2, we can regard the method
as either a clue to propose and examine inequalities like V1/V2 > 1/2
where V1 and V2 are the variables, or we study what happens when V1
and/or V2 have "slowly varying dimensional 'constants'" and get similar
inequalities either in V1 and V2 or in their DTs or DLs.
One surprise in this process is that a few variables give 0 in their
partial derivatives and occasionally there's a factor of 3. Frequency,
for example, with dimension T^(-1), has a DT partial derivative of 0,
while moment of inertia with dimensions ML^2 has a DT partial
derivative of 0, and surface tension has a TL of 0, while curvature has
a DT of 0. Power has a DT factor of 3 (I haven't found any others in
mechanics so far).
Variables with 0 or 3 as factors may or may not be indicative of a
paradox or anomalous problem. Frequency, for example, is what
allegedly makes quantum theory different from GR and classical physics,
so its 0 partial derivative reminds us that it has a "singularity" at T
= 0 and blows up near it if we regard it as approximated by
non-integers there. Curvature is what allegedly makes GR different
from classical physics (and early quantum theory), so its 0 DL partial
reminds us of a similar spatial "singularity at 0 since it has
dimensions L^(-1). On the other hand, I don't think that moment of
inertia or surface tension are necessarily anomalous despite their 0
partials and in fact time (T) commonly appears in the denominator of
mechanical variables without seeming to cause much trouble at 0 (though
arguably we should look into it).
This reminds me of a TV documentary that I saw recently regarding large
methane bubbles sinking ships in the Bermuda triangle region.
Engineers proved that methane bubbles (for example produced
occasionally from the earth and rising in the ocean) can easily sink a
large ship and even cause aircraft altimeters to have reverse readings!
So surface tension and viscosity and other hydrodynamic/aerodynamic
variables have some unexplored or only recently explored properties
that may have critical importance in transportation.
Osher Doctorow
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