Re: [Phys-l] The "why" questions

[curtis osterhoudt <flutzpah@yahoo.com>]

1. A wave is a wave is a wave. The K-K relations apply to them all, from 
acoustics to E&M to (we're assured) gravity. K-K applies because the functions 
describing these things are analytic in (at least) half of the Argand plane. 


2. Dispersion is *absolutely* key in the K-K relations, as they explicitly state 
relations between the real and imaginary parts (and therefore the "in-phase" and 
"quadrature" parts) of the functions. If you know the complete behavior of the 
real part you can find the imaginary behavior, and vice-versa. Or, if you know a 
suitable combination of these parts on a total domain which can be mapped to the 
upper-half of the Argand plane, everything is known (along with the knowledge of 
the analyticity of the functions). 


3. Lagrangian/Hamiltonian processes can absolutely have the K-K relations 
applied (viz. adiabatic approximations in optics, acoustics, etc.), so long as 
the processes in question are well-approximated by analytic functions. SHOs are 
an obvious example, but through the magic of Green's functions almost anything 
can be built up---very carefully---from appropriately shifted basis functions. 
It often gets very tricky when mode-conversions and attenuations or gains are 
incorporated, though. 


 /**************************************
"The four points of the compass be logic, knowledge, wisdom and the unknown. 
Some do bow in that final direction. Others advance upon it. To bow before the 
one is to lose sight of the three. I may submit to the unknown, but never to the 
unknowable." ~~Roger Zelazny, in "Lord of Light"
***************************************/




________________________________
From: Stefan Jeglinski <jeglin@4pi.com>
To: Forum for Physics Educators <phys-l@carnot.physics.buffalo.edu>
Sent: Tue, November 30, 2010 10:07:52 AM
Subject: Re: [Phys-l] The "why" questions

>  > I am not sure that I could argue persuasively because I never
> >  believed I quite got it back in the day, but what about this:
>  > http://en.wikipedia.org/wiki/Kramers-Kronig_relation
>
> That's quite an open-ended question, but let me guess that
> the intent was to ask whether the K-K relation is asymmetric
> with respect to time.

I'm not sure :-)

I have 3 different competing notions going on in my head and I can't 
find a way to make them come together (or remain separate if need be).

My original intent with the KK comment was to at least refute the 
notion that physics "never deals explicitly and directly with these 
ideas" [regarding causality].

1. Hand in hand, discussions of KK always are connected to causality. 
I haven't in a while tried going thru the detailed explanations of 
this and haven't here either, so I'm a bit at a disadvantage, but the 
discussions seem restricted to E&M (or at least most appropriate 
there). I don't know if that is an artificial restriction or not. Why 
is there not some KK-like relationship in Lagrangian/Hamiltonian 
mechanics, for example?

2. Like some others, I am squarely in the camp of F(t) = ma(t), where 
t is t. To me, any non-philosophical discussion of causality would 
require consideration of integral formulations of F(t) ~ ma(t-t') or 
a(t) ~ (1/m)F(t-t').

3. But speaking of which, I previously posed a question about 
convolution. It was pointed out that convolution merely expresses the 
idea that an output is a
superposition of impulse responses. Considering linear systems alone, 
it was also pointed out that some relationships are of the form

    (a) f(t) ~ y(t)

while others are more generally of the form

    (b) f(t) ~ Integral[h(t')y(t-t')dt']

These are the same only to the extent that h can be expressed as a 
delta function. JD gave the example of Ohm's Law being in camp (a) 
whereas anything with a capacitor in it requires camp (b). The 
obvious difference between these 2 entities (resistor/capacitor) is a 
phase difference between voltage and current. And interestingly, we 
are again led to a similar painful discussion! To wit, does voltage 
"cause" current or is it the movement of charge that "causes" a 
potential difference to be created? I =want= to identify this with 
the F(t) = ma(t) discussion but I'm not so sure it's as clear.

Somehow I feel there is a more formal connection between these 3 
items. My intuition tells me that dispersion is at least in part the 
key.


Stefan Jeglinski

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