Natural Science Forum / Physics / General Physics / May 2007

See the first post.

I guess I don't make a sharp distinction between reading a book and studying
from it.  There are certainly degrees of intensity with which one can
explore the material in a book.  I certainly could benefit from working
more exercises.  OTOH, I am incapable of reading mathematical physics
without (believing that I am) understanding.  My brain just shuts off if I
don't get something.  I can't read the subsequent material.  

For example, I was reading a book on elasticity, and I hadn't intuitively
grasped what is meant by stress.  I could read the words, and follow the
equations, but there was something that hadn't clicked.  I figured that it
would come to me if I just kept reading, so I went onto the next section.
For some reason I just couldn't understand the stuff.  I tried to go
through the derivations, and it just wasn't clicking.  Then I went back an
thought threw the introductory discussion again and finally figured out the
part I had been missing.  When I returned to the math I had been stumbling
over, I realized it was trivial algebra and trig.

If I had seen the same equations out of context, they would have been
obvious to me.  For some reason, my brain simply refuses to move past
something I don't get.  What makes that all the more difficult is that my
standard of understanding is typically far more demanding than that of
others.  That's why I hate the presence of the permittivity constant in
E&M.  I see this epsilon_o (epsilon sub omicron) in the expression, and I
want to know what it means.  It's bullshit!  Either explain it before you
use it, or get it out of the #@!$%^ expression!

If you are talking about variational method, I have to say, I find it to be
potentially misleading.  For example, Symon's comment on page 366: "Since
Lagrange's equations have been derived from Newton's equations of motion,
they do not represent a new physical theory, but merely a different but
equivalent way of expressing the same laws of motion."  In itself, the
statement is correct.  OTOH, there are subtle differences between variation
and differentiation.  When used correctly, variation is an operation on a
functional, not on a function.

I've been trying to find the specific example that I recognized years ago
where the generalized momentum and position lead to different results than
would their Newtonian counterparts.  It was something like the derivative
of position with respect to acceleration is zero in analytical dynamics,
whereas it is not, in general in Newtonian dynamics.   I remember it was a
derivative that a person would not typically perform directly, but it arose
in a derivation.

I tried that school thing a few times.  I always felt like I had a choice
between passing a course or learning something meaningful.

$20.  Years ago.

I was really wondering if there is anything /that/ exceptional about
Goldstein's book.  I have plenty of books on variational methods.  Right
now, I think the thing for me to do is finish Lemons's book and to spin up
my own formal development of Hamilton's equations.

You've noticed that too, eh?  Actually, as I see things, there are two kind
of not understanding.  There is the mundane form which is most common.  In
that form, the person simply fails to appreciate the implications of what
he or she has reiterated by memory or expressed in the form of symbol
manipulation.  In the second form the failure to understand is profound.
In this form the person knows that he or she does not understand nature
through the current theoretical framework because the current framework is
incomplete and, to some extent, internally inconsistent.

Concepts from variational dynamics are probably useful in understanding
classical field theory.  It's not a very big stretch to arrive at ideas
related to QED from there.  Schrödinger introduced QM beginning
with "Derivation of the fundamental idea of wave mechanics from Hamilton's
analogy between ordinary mechanics and geometrical optics."

I freely admit that I am not extremely well versed in QM.  I do, however,
have a basic understanding of the ideas, and have been able to read the
mathematical formalism with the kind of understanding that enables me to
verify the derivations.  From what I know of QM, it seems that most of it
is simply the application of abstractions taken from classical mechanics to
a non-classical domain.  That is to say, for example, Schrödinger's wave
equation is a second order partial differential equation relating potential
and kinetic energy.  The distinctions between QM and classical mechanics
are generally rules restricting the applicability of certain concepts of CM
in QM, or rules changing the interpretation of such values as wave
amplitude.

Fluid dynamics has a wealth of material to build a good foundation with.
You can throw in Riemannian geometry, some really mind-bending
transformations, all of the basic vector calculus, tensors, stress, strain,
angular momentum, oscillators, resonant cavities, wave propagation.
Thermo-dynamics fits right in when you start playing with changes in
pressure.  The list goes on, and on.

E&M should be learned from the relativistic perspective early on.

Which fad?

There are many different ways to express and/or learn any of these concepts.
For the way I think, if I can't draw a picture that, in some meaningful
way, corresponds to physical reality, I don't feel as if I understand the
physics.

<snip>

I really like the idea of the first course(s)- it's probably a 1-year
class, when all is said and done.  For (b), I would substitute other
topics: phase transitions, scattering, but definitely move into
phase-space and state space.  Maybe quantization of the action near the end.

The 'standard' physics curriculum is due for an overhaul.