Proof and meaning of Noether's theorem Most books I've consulted on this matter give the following statement of Noether's theorem:  If a Lagrangian has a continuous symmetry then there exists a current associated with that symmetry that is conserved when the equations of motion are satisfied.  In my notes, the theorem is stated rather differently: if the action $S[\Phi]$ is invariant to the first order with respect to infinitesimal trasformations $\Phi\to \Phi+\theta\Delta\Phi$ if and only if the parameter $\theta$ is global, then there exists a local conserved current $j^\mu(x)$. There is something I am not sure about. By the principle of stationary action, the action should be invariant (to first order) under a transformation $\Phi(x)\to \Phi(x)+\Delta\Phi(x)$; this is true for on-shell fields, that is, for fields that actually solve the resulting equations of motion. Therefore, on-shell, it should also be invariant when $\theta$ is a global parameter and so any action $S[\Phi]$ is trivially invariant under a transformation of the form $\Phi(x)\to \Phi(x)+\theta\Delta\Phi(x)$. In this version of Noether's theorem, should I assume the invariance to also hold off-shell, for any possible field $\Phi(x)$?
On-shell and off-shell transformations in Noether's theorem For any transformation of the fields,  $$\varphi\to\varphi'=\varphi+\delta\varphi$$ the change in the Lagrangian can be written as  $$\delta\mathcal L = \text{EoM} + \partial_\mu j^\mu\tag{1}$$where "EoM" represents the equations of motion (Euler-Lagrange equations) and all other terms can be written as a total derivative of some function $j^\mu$, which is a known function in terms of the Lagrangian.   I would like to distinguish the different realizations of transformations. Let's assume that the transformation (1) leaves the action invariant, $\delta S=0$.    $\delta\mathcal L=0$   EoM $=0$, "on-shell": Noether current is conserved, $\partial_\mu j^\mu=0$. EoM $=\partial_\mu b^\mu\neq0$, "off-shell": modified Noether current $J^\mu = j^\mu+b^\mu$ is conserved, $\partial_\mu J^\mu=0$.   $\delta\mathcal L =\partial_\mu a^\mu \neq 0$, "quasi-symmetry"   EoM $=0$, "on-shell": modified Noether current $J^\mu = j^\mu-a^\mu$ is conserved, $\partial_\mu J^\mu=0$.  EoM $=\partial_\mu b^\mu\neq0$, "off-shell": modified Noether current $J^\mu = j^\mu-a^\mu+b^\mu$ is conserved, $\partial_\mu J^\mu=0$.     Is this listing correct?      What roles do the terms "on/off-shell" and "(quasi) symmetry" play in Noether's theorem?   Related: , , , , .
Why sum of two little "o" notation is equal little "o" notation from sum? Why sum of two little "o" notation is equal little "o" notation from sum?  $o( f(n) ) + o( g(n) ) = o( f(n) + g(n) ) ?$  For example:   $f(n) = n^3$  $g(n) = 1/n$   so    $o(f(n)) = n^2$  $o(g(n)) = 1/n^2$   and    $o( f(n) ) + o( g(n) ) = n^2 + 1/n^2$ $o( f(n) + g(n) ) = n^2$   Of course, I could write it like   $o( f(n) ) + o( g(n) ) = n^2 + o( g(n) )$ $o( f(n) + g(n) ) = n^2 + o( g(n) )$   My question is why? I don't understand it, because in first we always get two parameters. So, if we have a large pipe which is constricted in the middle, the fluid in the middle section  will move faster. Why is it so? What dictates that liquid that an X amount of it must pass on the other side regardless of the size of the encasement?  The only reason I could think about is because that the initial pressure which causes the movement somehow is being transferred to a smaller area, but that makes no sense. I have a problem with the endfloat package and unicode characters.  When I compile the following, everything works as expected:  \documentclass{article} \usepackage[utf8]{inputenc} \usepackage[T1]{fontenc} \usepackage{graphicx}  \begin{document}  \begin{figure} \caption{Ö} \end{figure}  \end{document}   However, when I also load the endfloat package (\usepackage{endfloat}), I get the following error message:  ! Package inputenc Error: Unicode char \u8:ÖÖ not set up for use with LaTeX.   This could be a bug, but my suspicion is that I'm making a mistake somewhere.  I'm hardly the first person to use special characters with the endfloat package.  However, the above example is so simple, I don't see where I could have made a mistake.  Any advice is appreciated.  Edit: The file is of course encoded using utf-8. Using the itemize environment inside a caption Should flagging a question for closure prevent close-voting? I'm asking this because my HDD (Hard Disk) has 5 years, and ultimately (this year) it gave me many headaches. Many moths ago I has Windows 7 (and 10 T.P. since april) and all was good. But after a moment, instantly, all gets reversed: veeery slow, and actions which will take a tiny time, at these moment the time seems frozen (for example, instead a 1 minute boot, it turned to 5-8 minutes boot). I formatted and installed Windows every 2 months because it was the time which runs good before it gets slooww. Seeing the abuse of privacity in W10, I changed to Ubuntu 15.04 and the things lasted more in time... But again, HDD error. Click, trying to read, click, trying to read... And the upper bar (title of window/buttons of volume, language, wifi, etc.) disappeared. The left bar (where are the app's icons) was empty and the file explorer said the resource wasn't available (or something like these error). I restarted and nothing happens. I had to press CTRL+D to boot-up Ubuntu, but after hours... the same error (can't do anything except moving the mouse/cursor).  Welcome to the routine: copy files with live-cd to a pendrive, wipe HDD and install all again. And the things was calmed again... and the errors comes again. And after 2 weeks, I had to do the same. Now my PC works fine, but yesterday I heard a few "clicks" from the HDD.  Now, I remembered this  and I came to Disks and the HDD shows a new value; normally, the problematic number was "pending sectors", always changing, but now, in "irrecoverable sectors", the number is different of zero.  I never had "reallocated sectors". Now, neither. And now, while I write this lines, I've 5 "pending s." and 9 "irrecoverable s.". If this helps, the "read error rate" is 42, 5 for "write error rate".      Very well!! While I was writing "write error rate", I heard 2   clicks. And while writing this, one click more.   And 123 "CRC error rate". So, with this information: is my HDD dying? It'll last days or months? I think this problem will kill the disk in a not very far time.  The drive is a Western Digital Green 1TB-WDC WD10EARS-00Y5B1 (80.00A80) S.N.: WD-WCAV5J301610  PS: When I boot with CTRL+D, many times the system is in "RO mode"; I can't save/download files, Unity doesn't find apps when I write their name {click again, now, while I write the text} and, after a few hours, a crash like the lines above How to get second last page as a number without hyperref? Can I get UAC prompt for user from batch file? Can I get UAC prompt for user from batch file?  For example I need put some keys to registry under CurrentControlSet from batch file by     reg import path\to\my.reg   PS. runas is not a solution as my customers uses different Windows localisation and I can not guess which name used for Administrator account (this can be Администратор)...  And script don't know administrator password.  And UAC dialog box does not require password entering... Optional Android notifications for reputation changes I would like to get a notification if my reputation count has changed. I think there is already some internal API for that. (I can see that changes in the menu bar in the top).  I'm sure that some people would not like that so that should be optional. Get the client IP address using PHP calculator's angle answer for trig ratios that can work in more than 1 quadrant on the unit circle Why does the calculator do a cc (counterclockwise) rotation for positive trig ratios instead of clockwise,  and a clockwise rotation for negative sine &amp; tan instead of cc and a counterclockwise rotation for negative cos ratios instead of a clockwise   ie. in degree mode  $\cos^{-1}(-5/12)=114.62$  $\sin^{-1}(-5/12)=-24.62$  $\tan^{-1}(-5/12)=-22.61$  Is it maybe picking the value that involves the least amount of computing power? or is it a matter of convention? or am I overlooking something? I read many places that the rule of thumb for swap space is to double the amount of physical RAM. However, 32 GB does seem a LOT. Do I need that much? Do I need it at all with this high amount of physical RAM? Sometimes when you're downloading an image and the connection breaks mid stream, you are left with a half downloaded image. If you try to view it you get the upper part of the image and the bottom part is usually coloured grey or green or some other color.  In other words, it is corrupted.  Is there a way to check whether the image is damaged in that way or otherwise corrupted? I am currently writing my thesis and need to include a bunch of email conversions. I am using classicthesis. I wonder, how you would format stacked conversions. Moreover, how would you deal with the email header, signatures and line breaks: why do I need to add \\ all the time? This is the configuration I am currently experimenting with:  \documentclass[ oneside, openright, titlepage, dottedtoc, numbers=noenddot, headinclude, footinclude=true, cleardoublepage=empty, abstractoff, BCOR=5mm, paper=a4, fontsize=11pt, ngerman, american ] {scrreprt}  \PassOptionsToPackage{utf8}{inputenc} \usepackage{inputenc} \usepackage{blindtext}  \usepackage{framed}  % Vertical line that highlights quotations in an email. % Use \begin{leftbar} and \end{leftbar} environment. % Width of the line = 0.4pt % Space between the line and the text = 10pt % Requires \usepackage{framed} \renewenvironment{leftbar}{%   \def\FrameCommand{\vrule width 0.4pt \hspace{10pt}}%   \MakeFramed {\advance\hsize-\width \FrameRestore}}% {\endMakeFramed}  \usepackage{nicefrac} \usepackage[english]{babel} \usepackage[pdfspacing]{classicthesis}  \begin{document} \selectlanguage{ngerman} % american ngerman % Disable paragraph indentation. \setlength{\parindent}{0pt}     \begin{verbatim} From: john.doe@yahoo.com Subject: Your inquiry from 20042012 Date: Thu, 3 May 2012 11:34:54 +0200 To: Jane Doe &lt;jane.doe@hotmail.com&gt; \end{verbatim}  Jane Doe wrote: \begin{leftbar} \begin{leftbar} \begin{leftbar} \begin{leftbar} There are a couple of questions I like to ask. Please forward them if you cannot do so. \end{leftbar} I hope I can answer them after holidays. \end{leftbar} I am back from holidays now and will look through your questionaire the next days. \end{leftbar} Thank you, Mr. Doe - I am awaiting your answers. \end{leftbar}  Dear Mrs. Doe,\\ \ Lorem ipsum dolor sit amet, consectetur adipiscing elit. Ut molestie placerat pharetra. In quam leo, tincidunt ut consequat eget, malesuada sit amet urna. Morbi et interdum arcu. Nullam nisi nunc, ullamcorper vitae pulvinar eu, suscipit ut augue.\\ \ Donec risus mi, accumsan ac interdum id, molestie sed mauris. Suspendisse porttitor luctus justo in ullamcorper. Proin in neque magna. Mauris quis arcu eget enim mattis pharetra vitae a dolor. Nam aliquam consectetur mollis. Ut at sem quis dolor faucibus fringilla. Quisque cursus vehicula odio, vitae placerat ipsum dapibus id.\\ \ Best regards\\ John Doe\\ \ (See attached file: Questions\_Doe.doc)  \begin{verbatim} _______________________________________________________ Administration New York John Doe Mainstr. 3 11111 New York  T:        (011) 212-4301 F:            (011) 212-954301 M:         john.doe@yahoo.com \end{verbatim}  \begin{leftbar} My question I looking forward to reading the answers to my questions:\\ \ 1. Is the world analog or digital? \end{leftbar} digital  \begin{leftbar} 2. How long does the world exist?  \end{leftbar} analog since 1984, digital since 1999  \end{document}     This is what the above latex example looks like.    I also saw  but I do not want to use &gt; or colors. My settings are inspired by this  and this . Are there solutions that can limit the CPU usage of a process? I am using an old laptop and doing heavy processing that needs high CPU usage for a long time (~30mins - 2hours). When the process puts heavy load on CPU for a long time, CPU overheats and computer shuts down. I don't want to use a cooler/stand because I am carrying this laptop around and working outside home.  Is there a way/application that I can limit CPU usage of a process in Windows?  Note: Setting the priority of the process from task manager does not work.  Note 2: Fans are clean. What's the problem with my Young slits experiment? My Young's slits experiment doesn’t work. I mean I tried to see wave effect of light but I didn’t see any thing expect particles! I’ve sent you a pic of my laser and a short video of the experiment. Is there any problem you think? Does math have an incorrect notation / syntax? I don't mean writing misaligned notation (google), but when you take something like a number to powers to powers to powers, $${{2^2}^2}^3$$ (I was told this is incorrect notation by a teacher). Is it really incorrect, or does it just need to be simplified with parentheses? Do people write maths like this?  a radical expression with the root being a radical expression? $$\sqrt[\sqrt{2^3}]{2}$$ Is there a command to delete all the files in a directory that haven't been modified in N days?  I need to clean up some old logs. I know there are some similar posts on adding review queues to the mobile web page. However, this is regarding the Stack Exchange Android app. I really would like to review posts and edits from the Android app, however this isn't included in the app yet. I'm not sure how it would be implemented; maybe there could be a button or something for review. Could we please see this? $\LaTeX$ and Mathematica