Linearity;Nonlinearity and why it needs math to make precise

[Archimedes Plutonium]

WE know the math definition of precision for linear Y = mX + b

But that form does not transfer over to physics where they talk about
linearity and nonlinearity. So, what are these physicists talking
about?

One would think that the journals devoted to nonlinear physics would
define precisely what nonlinear physics is? Surprize, they run and hide

Of course the Maxwell Equations, Dirac Eq. Schroedinger Equation are
Linear. But is the Heisenberg Uncertainty Principle a linear relation?
Most would say yes. But is it really, when considering that the
definition of linear in physics is not well defined and precise as what
the rigors of math defining are.

Those readers who have followed me for 2 years know that I am horribly
precise. And the reason the 4 Color Mapping problem is a 2 color
mapping is because math people crumble when it comes to precision. And
the reason I proved FLT is false is because the definition of 'finite'
was lacking in math. But here I am faced with ILL-defining in physics
and I am asking for the math people to come over and help these poor
physics people.

Linear in physics

When physicists talk about linearity such as say the Maxwell
Equations

get a wave equation by solving the differential equation. Solve for
answer

and you get one or more solutions. Generally more than one, and usually
a set

of functions that satisfy equation. Some even have special names such
as the

Bessel functions or the Legendre functions. Now, here we distinguish

linearity from nonlinearity. Linear equations have the property that
the sum

of two or more solutions is also a solution.

So you come to the idea that an arbitrary solution or any solution to

linear equation can be written as teh sum of other solutions. Here, we
see

that the superposition principle is key to linearity. And linearity is
some

kind of vector space-- space where each axis corresponds to a solution.
Or,

linearity is a kind of Fourier Series--- sum of sin and cosine

Another way of seeing Linearity in Physics, perhaps easier, is to
look

at the system as a black box and it has an input and an output. Now

if you put in a sin wave or a signal, then the output is proportional
to the

input. Just multiply by some number, like an amplifier, and the wave is

not distorted in linearity. Put function in, then the output looks like

what the input was, only changes is amplification.

Physics needs math precision as to the definition of Linearity vice

NonLinearity

[Tom Potter]

In <4i1qdk$s...@dartvax.dartmouth.edu>
Archimedes...@dartmouth.edu (Archimedes Plutonium) writes:

>>>snip<<<

> Physics needs math precision as to the definition of Linearity vice
>NonLinearity

It seems to me that nonlinearity is when two quantities are related by
an exponent which is not equal to one or zero.

In other words:

linear Y = A * X^N where N = 0 or 1

nonlinear Y = A * X^N where N <> 0 or 1

What do you think?

[RAFAEL M RUBIO]

In article <4i1qdk$s...@dartvax.dartmouth.edu> Archimedes...@dartmouth.edu (Archimedes Plutonium) writes:
>WE know the math definition of precision for linear Y = mX + b
>
>But that form does not transfer over to physics where they talk about
>linearity and nonlinearity. So, what are these physicists talking
>about?
>
>One would think that the journals devoted to nonlinear physics would
>define precisely what nonlinear physics is? Surprize, they run and hide
>

The defintion of a linear system in physics is any system where
the principle of superposition holds. This is basic knowledge
in physics and you will find it in sophomore/junior level
undergraduate physics courses. There is no reason for
physicists to need a review when reading nonlinear journals.

> (Section deleted on your apparent "discovery" of precise definition)

>
> Another way of seeing Linearity in Physics, perhaps easier, is to
>look at the system as a black box and it has an input and an output.
>Now if you put in a sin wave or a signal, then the output is
>proportional to the input. Just multiply by some number, like an
>amplifier, and the wave is not distorted in linearity. Put function in,
>then the output looks like what the input was, only changes is
>amplification.
>

> Physics needs math precision as to the definition of Linearity vice
>

Your above defintion of linearity is myopic and unpratical. However,
your "discovery" was correct. There really is no lack of precision
of which you speak but only a minor oversight on your part.

Now if you say that physics literature is difficult to pick up and
read without a degree in the field. That I will give you.

Rafael
rar...@nmsu.edu

[Zamin Iqbal]

Morning everyone.

I'm looking for some kind of characterisation of the elements of SO(n) for n>2. For n=2 we can write an element as a matrix in terms of cosines and sines, but what about higher dimensions? Since an axis corresponds to an eigenvalue 1, is it true that elements of SO(n) for n odd have to have an invariant axis, but elements of SO(n) for n even do not?

Please send replies to this email address.

Thanks

Zam

[Bruce Dean]

Zamin Iqbal wrote:
>
> Morning everyone.
>
> I'm looking for some kind of characterisation of the elements of SO(n) for n>2. For n=2 we can write an element as a matrix in terms of cosines
and sines, but
>

> Please send replies to this email address.
>
> Thanks
>
> Zam

Interesting problem and a natural question to ask. You may want to
check out some of the literature on Clifford Algebra's. Using this
approach you may completely characterize the orthogonal groups: O(p,q),
i.e the group of transformations leaving invariant the metric with
signature = diag(1,1,...1,-1,-1,...,-1); having p (+1)'s and q (-1)'s.

This stuff is a lot of fun to play with...best wishes.
Sincerely,

Bruce Dean
Mathematics Dept. WVU
Morgantown, WV 26506

[Bruce Dean]