Gauge transformation in the Lorentz force Lagrangian.

November 2, 2020 Uncategorized , , ,

Problem: Lorentz force gauge transformation.

Show that the gauge transformation $$A \rightarrow A + \grad \psi$$ applied to the Lorentz force Lagrangian
\label{eqn:gaugeLorentzSTA:20}
L = \inv{2} m v^2 + q A \cdot v/c,

does not change the equations of motion.

The gauge transformed Lagrangian is
\label{eqn:gaugeLorentzSTA:40}
L = \inv{2} m v^2 + q A \cdot v/c + \frac{q v}{c} \cdot \grad \phi.

We know that the Lorentz force equations are obtained from the first two terms, so need only consider the effects of the new $$\phi$$ dependent term on the action. First observe that
\label{eqn:gaugeLorentzSTA:60}
=
\frac{dx^\mu}{d\tau} \PD{x^\mu}{\phi}
=
\frac{d \phi}{d\tau}.

This means that the action is transformed to
\label{eqn:gaugeLorentzSTA:80}
S
\rightarrow S + \frac{q}{c} \int d\tau \frac{d\phi}{d\tau}
= S + \frac{q}{c} \evalbar{\phi}{\Delta \tau}.

As the action is evaluated over a fixed interval, the gauge transformation only changes the action by a constant, so the equations of motion are unchanged.

References

Editing a book by tearing out pages!

October 25, 2020 Uncategorized ,

I’ve been editing my classical mechanics notes compilation, which doesn’t yet justify being called a book.  Here’s the editing process in action:

I’ve purged about 120 pages, and wrote 16 new pages (covariant Lagrangian and Lorentz force material) to replace portions of some of the gutted material.  The stack beside the book is about 3/8″ thick of ripped out pages.  I ripped them out so that I could see the remainder more easily.

There’s still a lot to purge and rewrite, but I’m now down to a more manageable 347 pages, which is a good start.  Next up will be the material related to Lagrangian densities for field equations (wave equations, Maxwell’s equation, …)

Replacing some wall sconces. Electrical surprise!

October 3, 2020 Uncategorized , , ,

We had some wall sconces in the second floor hallway.  One of them sputtered and we didn’t trust it.  Sofia found some nice replacements a few weeks ago, and today was installation day.

I was rather surprised when I took off the old light and found a wiring hairball.  Here’s a picture after taking off (all but one of) the merretts:

I’d never seen anything like this, not even at my last house, where the previous owner (Mr. C) never found a ground wire that he didn’t like hanging loose.  You can’t really see into the back of the box in that picture, but all the wires are loose (no wiring clips securing the wires to the back of the box). none of the sheathing has been removed, and all the connections were jammed into a single tight cubic inch in an impossibly tangled mess.  The grounds were all tied, but Mr C would have been proud, as none of them were connected to the ground screws at the back of the box.  You can see things a bit better after a bit of unraveling:

Of these wires, I hadn’t yet identified which was switch, which went to the second sconce, and which was the supply.  I’m not sure about other locales, but the Ontario electrical handbook specifies that white is hot at the switch, so looking at this, I think the guess should be that the switch wire is the top left (i.e. when the switch is on, we have supply through the white, to and through the switch, and back through the black where it will power the load.)  That didn’t make sense since the switch is on the right down about a foot.  The only logical wire for the switch would be that bottom one.  I decoupled everything so that I could test the lines (carefully) with the power back on, warning everybody in the house to keep clear:

My tester showed that the top left was the supply, leaving the top right and the left as candidates for the switch and the other sconce.  A couple more trips to the breaker box, and some temporary connections verified what was what, and I was ready to start reconnecting things.  First step was stripping the sheathing off the wires.   I couldn’t use a standard stripping tool, since everything was already in the box, but had to carefully do that with a knife:

I’d never seen anybody not take the sheathing off, which was the major source of the hairball wiring.  This wire was either 12 gauge or just really old, but it was very stiff and hard to handle.  With all the sheathing still on, whoever wired this up originally must have had a hell of a job.  I got the switch’s white connected up to the supply black, and put in some clamps at the back of the box and secured the grounds all physically to the box.

I left one ground long to potentially connect to the new light, but it turned out that the ground wire on the new light was super long, so I ended up trimming that one back and direct connecting it to the back off the box with all the rest, instead of using a merrett for the ground.

Special bonus.

It seemed prudent to open up the switch too, and found that the ghost of Mr C was haunting that too.  Check out the nice floating ground lurking in the switch:

This single pole switch doesn’t have a ground screw, but it just seems really sloppy not to connect that ground to the back of the box.  This wire was also not clamped, so I did that too, and put in a brand new switch while I was at it.

After:

Small update to “Basic Statistical Mechanics” is now live.

August 8, 2020 Uncategorized

A new version of these notes is now posted, available on amazon, leanpub, and as a free pdf:

phy452.V0.1.12.pdf, Wed Aug 5, 2020 (commit 7bbcdf66b26e950fa01ae6cbae86f987bc2c8d49)

• Fix hyphens in listing, typos in bio.
• Remove appendix part so that the index and bib aren’t grouped with the appendix.
• Tweak the preface and backcover
• Group intro probability text together, and expand on probability distribution definition.
• Remove singlton part heading so that chapters are the highest level.
• Fix pdfbookmarks for contents and list of figures (so that they don’t show up under the preface)
• Streamline FrontBack specialization.

These are mostly cosmetic changes, where my primary objective was to correct the bash listing that shows the reader how to make their own git clone of the book text.

Leanpub editions of my books.

August 5, 2020 Uncategorized ,

I’d had a leanpub version of my geometric algebra book available for a while and have now added editions of all my older class notes compilations that I have on amazon.  My complete leanpub selection now looks like:

I believe that leanpub essentially provides a pdf to the purchaser (I haven’t tried buying a copy to verify), and I give the pdfs away for free, so you (and I) might ask why somebody would opt to buy such a copy?

There are a few possible reasons that I can think of:

1. Many of the leanpub purchases have been above the minimum price, so at least some of the purchasers are compensating proportionally to their personal valuation of the material, and aren’t strictly trying to buy for the minimum price.
2. A leanpub purchase is subscription like.  Anybody that purchases a copy will automatically receive any updates made without having to check for a new version manually.
3. There is a per-book forum available for each of the books (if the author enables it.)  I didn’t realize that feature was available, and have now enabled the forum for my geometric algebra book.  I’ve also enabled a forum for each of the class notes compilations as I configured them.
4. The purchaser did not know that I also offer the pdf for free, and found the title in leanpub search, not through my website where I make that obvious.

I’ve been putting all my leanpub proceeds into my kiva loan portfolio, so if somebody had the bad luck to buy a copy of my book because of (4) above, I don’t feel very guilty about it.