Variational Methods Discussion
from The Tesla List
Updated 7-29-06
Date: Thu, 20 Apr 2006 18:48:15
-0600 -Subject: Variational Methods - Original
poster: Jared E Dwarshuis
Hi Gerry
Variational methods are the calculus of variation. Extremum problems such as the
shortest path between two points or, the largest container from a given piece of
sheet metal, are examples of extremum problems.
The Lagrange is one of the variational methods used to examine the extremum of
potential energy and kinetic energy. (or stored energy in a capacitor and moving
charge in the inductor)
One could most likely adapt an already worked solution borrowed from mechanics
to nail down the equations of "frequency splitting". -
Sincerely: Jared Dwarshuis
Date: Fri, 21 Apr 2006 14:31:36
-0600 - Subject: Variational Methods -
Original
poster: Jared E Dwarshuis
We dont need Green, Maxwell or Schwinger to solve this. We only need to write
our equations correctly and be - oh - so - carefull – in applying the Lagrange.
None of the steps are beyond the capability of someone who has had linear and
diff-eq
The setup gets messy, lots of places to drop the ball. So it is best to have
someone reality check your work as you go along.
Simple in the grand scheme of things but it would take me several days of
playing with examples in textbooks before I would feel confident enough to begin
hammering out equations.
Reminds me of the physics problems with multiple weights and pulleys. God they
are a pain. But they can be done by mere mortals with a lot of patience.
My apology to Robert:
I used: La = T - V
Textbooks use: L = T - V
Didn't want to confuse inductance "L" with the Lagrange, which I labeled "La"
and yes.... T and V are energy.
Sincerely: Jared Dwarshuis
Date: Fri, 21 Apr 2006 19:49:00
-0600 - Subject: RE: Variational Methods -
Original poster: "Godfrey
Loudner"
Hello Jared
The Maxwell equations are equivalent to the least action principles, so we do
need Maxwell either directly or indirectly. If one views the tesla coil circuits
as purely lumped with small damping, then all you say can
be done with differential equations. In fact it all can be found in "Principles
of Electricity, Page & Adams, Chapter XV" and done without writing down a
Lagrangian. Even I can redevelop the chapter XV content
in terms of Lagrangians. I suppose if I were teaching from Page & Adams, I would
assign the task as an exercise. Well I never assign any problem I can't work
myself. But the chapter XV content is an approximation
because the currents are assumed to be uniform. The currents in a Tesla coil
secondary are not uniform. I was suggesting that in order to get a more accurate
picture of the secondary, perhaps least action principles
could be used. I think finding the associated Green's functions would be a
monumental task. Even if reasonable Green's functions could be found, then one
would probably have to resort to the voodoo techniques of
asymptotics, perturbation, or WKB approximations to make the process give up
interesting information. See "Advanced Mathematical Methods for Scientists and
Engineers I, Bender & Orszag".
Godfrey Loudner
Date: Sat, 22 Apr 2006 09:40:45
-0600 - Subject: Re: Variational Methods -
Original poster: Steve Conner
>One could most likely adapt an already worked
solution borrowed from
>mechanics to nail down the equations of
"frequency splitting"
Or you could just look them up on Richie Burnett's website.
Seriously, frequency splitting is perfectly well understood. It follows
naturally from the differential equations that describe the electrical behaviour
of the Tesla coil system.
It happens that I do have a background in mechanical engineering, and I do like
to think about it in terms of the problems we were taught in dynamics that use a
load of matrix equations to find the resonant frequencies of a structure. The
maths is pretty much identical. -
Steve Conner
Date: Sat, 22 Apr 2006 14:27:18
-0600 - Subject: Re: Variational Methods -
Original
poster: "Bob (R.A.) Jones"
Hi Jared
> Original poster: Jared E Dwarshuis
>
snip
> My apology to Robert:
I used: La = T - V Textbooks use: L = T - V
>
> Didn't want to confuse inductance "L" with the
Lagrange, which I
labeled "La"
and yes.... T and V are energy.
Yes I think you defined them as energy. But are they inductive and capacitive???
I would expect "total energy" = "capacitive" + "inductive" where "total
energy" is a constant under steady state conditions.
Robert (R. A.) Jones -
A1 Accounting, Inc., Fl -
407 649 6400
Date: Sun, 23 Apr 2006 14:09:21
-0600 - Subject: Variational Methods - Original
poster: Jared E Dwarshuis
Hi Robert:
Capacitive would be potential energy (T), and inductive would be kinetic (V). We
are not summing the total energy so the minus goes in the middle betweem T and
V.
For the math part you need a good textbook. I regrettably do not have a copy,
and I believe that it is out of print. But I particularly liked a Book written
by a guy named French, it was an M.I.T published book. Waves and Vibration? or
Vibration and Sound? Rats I can't remember the title anymore.
You deserve a good answer in plain talk. Here is what I have gathered about the
Lagrange.......
The Lagrange sort of steps back and says we can examine an entire system without
worying about specific points along the way. When the lagrange is used to
analyze Newtonian mechanics, we can toss out a lot
of vector analysis. WE are no longer interested in one point in reference to
another point or points.
The rough argument (in many instances), is that all the information we ever
really needed was contained in the energy expression. (so why wory about keeping
track of a pile of reference frames )
There are many kinematic problems solved with the lagrange that no one has ever
bothered to recast in the Newtonian mold. (Too much work, or nearly impossible
to get right).
The Lagrange is used to analyze both continuous systems (like rope resonance)
and discrete (or lumped) systems (like mass and spring)
We found that although a resonant transformer is a distributed system. It has
point solutions that are readily compatible with lumped models. Mechanical
solutions were the basic inspiration for our capacitively
coupled transformers. In some instances we took already worked solutions and
simply changed variables. In any case we used the Lagrange to see if a
particular solution was mathematically viable before building a prototype coil.
There are magnifier designs, just waiting to happen! -
Respectfully: Jared Dwarshuis