Noether’s Theorem: Proof + Where it Fails (Diffeomorphisms)

[Alfred Einstead]

On May 24, 4:52 pm, Alfred Einstead <federation2...@netzero.com>
wrote:
> I decided to give scribd out a try and (if it works out okay) to place
> all my notes, expositories, some of my unpublished research, etc. on there.

The expository named in the subject header has been uploaded to
Scribd, at

http://www.scribd.com/doc/144954476/Noether%E2%80%99s-Theorem-Proof-Where-it-Fails-Diffeomorphisms

In addition, the following has also be uploaded
The Helmholz Conditions and Field Equations
http://www.scribd.com/doc/144956010/The-Helmholz-Conditions-and-Field-Equations

Summary (of the Noether Theorem article):
There is a simple way to state and prove the Noether theorem that
allows you to see all the important features – the
(symmetric) stress tensor, the force law, the conservation laws ...
and most importantly: the places where the theorem
does not apply (i.e. where it actually breaks down), but in which the
theorem commonly and naively used.

So, this note is just as much about dispelling some folklore as it is
about proving a result.

This is a reformatted, reworked version of an article posted in
sci.physics.research some time around 2010 or 2011.

It is a work in progress (as seen by the highlighted notes and
incomplete reference list) and, time permitting, will be updated.

Summary (of the Helmholz Conditions article):
The Helmholz conditions determine when a dynamics has a formulation by
a Lagrangian. There is a version for mechanics, as well as a version
for field theory; the latter being more general. The field-theoretic
version of the Helmholz conditions is derived here.

This is a legacy article from my archive, which I may update, time
permitting.
– Mark Hopkins

A note on this:
> The content spans a large number of fields of mathematics,
> physics, computer science, logic and other fields in addition.

The "notes" are 50 notebooks, 200 pages each, each page translating to
about 10 pages in book/monograph/paper form. I won't be posting *all*
the announcements here.

The Newman-Penrose expository I also just uploaded was at 8 "reads"
before I even finished writing any announcement or summary of it to
post here -- about 30 seconds. (Now up to 10, as I'm writing this.)
That's some good database linkage they go going on at that site.

[Alfred Einstead]

On May 31, 4:44 pm, Alfred Einstead <federation2...@netzero.com>
wrote:
> The [Noether's Theorem] expository ... has been uploaded to Scribd...
> It is a work in progress ... and ... will be updated.

The site does version tracking and updating. All the articles
mentioned in previous posts have therefore been updated and will
continue to be, if they are listed as "works in progress".

> The "notes" [the articles are drawn from] are 50 notebooks, 200 pages each,

> each page translating to about 10 pages in book/monograph/paper form.

After removing the repeat coverage, and combining stuff, it could come
out to 10% of the raw total or 10,000 pages. What's on Scribd is just
the stuff on my machine (about 6GB). I haven't gotten to the notebooks
yet. :)

This week there will only be one announcement. I've uploaded the
following to Scribd:

(1) On Physical Lines of Force (Maxwell, 1861)
http://www.scribd.com/doc/145552238/On-Physics-Lines-of-Force-Maxwell
This is an annotated reproduction of the original, from Maxwell,
published in 4 parts in 1861-1862 in Philosophical Magazine. The
verbal text has not yet been double-entry validated.

The modern notational equivalents of key equations and quantities are
listed in footnotes, along with further descriptive commentary.

(2) A Dynamical Theory of the Electromagnetic Field (Maxwell, 1864)
http://www.scribd.com/doc/145552677/A-Dynamical-Theory-of-the-Electromagnetic-Field
This is an annotated reproduction of the original, from Maxwell,
published by the Royal Society in 1864-1865. The verbal text has been
independently double-entry validated.

The modern notational equivalents of the key equations and quantities
are also listed in footnotes. The table in section 70 is expanded to
include the implicit 21st-23rd equations and to list, alongside, the
modern renditions of the key items.

Both of these PDF's were actually recovered from my machine after my
hard drive crashed; the original files are gone and the PDFs will have
to be reformatted and reworked in the future.

Other articles uploaded include the following:

(3) Normal Forms
http://www.scribd.com/doc/145551174/Symplectic-Representation-of-the-Tardion-Normal-Forms

This is a reformatted, rewritten version of the articles below, which
I originally posted. In the process of discussing the spin-orbit
decomposition, a normal form for the symplectic leafs corresponding to
the spin non-zero tardion is derived. Out of this also comes the
Newton-Wigner position vector.

Originally: Re: The First Law and Angular Momentum, sci.physics
Part 1: 2009 December 11 14:07:12 -0800 (PST)
Part 2: 2009 December 11 14:41:33 -0800 (PST)

(4) The Lagrangian Method
http://www.scribd.com/doc/145550361/The-Lagrangian-Method

An interesting numerical method for dynamics foregoes the differential
equations of motion working, instead, directly with the action. Since
the object of a numerical evolution is to find a sequence of points
that approximates the motion of the system, the question comes down to
formulating the least action principle with repsect to a path defined
piecewise by a given interpolation. Assuming the interpolation is
fixed, this reduces to an optimization problem over the point
sequence, itself. Solving this yields a series of iteration equations
which can then be used to numerically evolve the system. Since the
interpolation is fixed, this produces a suboptimal solution, but one
optimal with respect to the constraint.

The process is illustrated with application to the simple harmonic
oscillator and the Kepler problem. A notable feature in the latter
application is that the iteration involves logarithms, rather than
polynomials!