I know that if we have a lagrangian such that  $$ L'=L+\frac{d}{dt}(f(q,t))$$ then the equation of motion will be the same for $L$ or $L'$.  But I would like to know if there is a proof of the opposite, ie :  If $L$ and $L'$ describe the same motion, then $$ L'=L+\frac{d}{dt}(f(q,t))$$  I don't know how to prove it?
So I was reading this:  and while the answers for the first question are good, nobody gave much attention to the second one. In fact, people only said that it can be proved without giving any proof or any.  So, if I have a Lagrangian and ADD an arbitrary function of $\dot{q}$, $q$ and $t$ in such a way that the equations of motion are the same, does this extra function MUST be a total time derivative?   EDIT Ok, I changed my question a little bit:  Question: If I have a function that obeys the Euler-Lagrange equation off-shell, this implies that my function is a time derivative? (This was used in Qmechanic's answer of this other question: , equation 14.)  Also, why people only talk about things that change the Lagrangian only by a total derivative? If this is not always the case that keeps the equation of motion the same, so why is it so important? And why in the two questions I posted about the same statement on Landau&amp;Lifshitz's mechanics book only consider this kind of change in the Lagrangian?
How to limit zoom level in OpenLayers 2.x I would like to have only 3 levels of zoom available (from level 6 to level 8).  demolayer2 = new OpenLayers.Layer.WMS( "abc...","http://localhost:8080/geoserver/gwc/service/wms", {layers: 'def...', transparent:"true", format: 'image/png', numZoomLevels: 3, minZoomLevel: 6 },{isBaseLayer:false},  { tileSize: new OpenLayers.Size(256,256)});  map.addLayer(demolayer2);   map.addControl(new OpenLayers.Control.LayerSwitcher({'div':OpenLayers.Util.getElement('layerswitcher')}));  map.zoomToExtent(new OpenLayers.Bounds(-4.601615515076983,39.8769407866263,-3.0527184873294764,41.16710732525929)); I have recently analysed my website with pagespeed addon on firebug. It suggested me to set expiration on CSS, JS and image files.  I am wondering how do i do this? So I tried a few different answers that I found here, and I feel that this is possibly a duplicate question, but everything I have tried from the other solutions hasn't worked.  Basically I am trying to bake a material to a texture that will be used in Unity, but every test gives me a black background image.  In the image below, the top image texture is the mask (black and white, white being the leaves), and the bottom image texture is the colour. SSH login with clear text password as a parameter? I need to login to a user that I've created on a remote host running Ubuntu. I can't use an ssh key because the ssh login will happen from a bash script ran within a server that I won't have access to (think continuous integration server like Bamboo).   I understand this isn't an ideal practice, but I want to either set the remote host to not ask for the password or be able to login with something like ssh --passsword foobar user@host, kind of like MySQL allows you to do for logins.   I'm not finding this in man ssh and  I'm open to any alternatives to getting around this issue. If I create just a basic hello.tex file an compile it using   pdflatex hello.tex   The pdflatex program is creating a plethora of file which is a little annoying. Currently it writes all those files about which I really don't care:  hello.aux, hello.log, hello.nav, hello.out, hello.snm, hello.toc   Is there a way to tell pdflatex to only write hello.pdf if the build is successful ? If not, can I specify the output directory of those files ? Maybe I could just write them in /tmp. How to print the name of missing files in a folder? Is it OK to call a professor by his first name when he/she signs emails by only first name? If a professor in a North American country presents his- or herself by his/her first name in email messages, does this mean that students can refer to him/her by his/her first name? Or is this generally not a good idea, unless the professor has explicitly mentioned that he/she can be referred to by first name? I've noticed that most professors, who prefer to be referred to more formally, do not sign their emails with just their first names, but usually initials or first and last name. Are there any references to Eru or the Valar in the Lord of the Rings movies? The closest references to Eru or the Valar in Peter Jackson's Lord of the Rings and The Hobbit movies I can think of are Gandalf's vague description of his resurrection ("I've been sent back... until my task is done.") and his later description of Aman to Pippin ("White shores and beyond, a far green country under a swift sunrise." -- as an aside, he's taunting Pippin here, since Hobbits suffer from the Gift Of Men) and the equally vague information viewers are given about where the elves are going when they leave Middle Earth.  Are there any more direct references I'm missing? Maybe a mention of Aule by a dwarf? Calculation of $\lim_{n\to \infty}\prod_{r=1}^{n} \left(1+\frac{r}{n}\right)^{\frac{1}{n}}$ How do I insert a line at the top of a text file using the command line? As you can see from the image below, I have an issue with the text alignment of extra long chapter names. How can I fix it so that the text will not go further left than the start of the chapter name? That is, I don't want any text to be underneath the "Appendix A" part.  I added that red line just to show how I'd like it aligned, if possible.    Here's my MWE:  \documentclass{book} \let\cleardoublepage\clearpage  \RequirePackage{titletoc} \RequirePackage{tikz} \RequirePackage[explicit]{titlesec} \RequirePackage{DejaVuSansCondensed} \RequirePackage[normalem]{ulem} \RequirePackage[   left=2.25in,   right=0.75in,   top=1.25in,   bottom=1.25in,   marginparwidth=1.75in,   marginparsep=.25in,   asymmetric]{geometry}  \renewcommand*\familydefault{\sfdefault} % Set default font  %%% Table of Contents ---------------------------------------------------------- \setcounter{tocdepth}{1} % Show only Chapters and Sections  %%% Change font/color/layout of TOC \makeatletter \titlecontents{chapter}[-2em] {\color{cyan}\bfseries\LARGE\addvspace{3mm}} {\makebox[4.4em][r]{\@chapapp}\contentslabel{-0.25em}\hspace{1.9em}} {} {\hspace{3em}\contentspage} \g@addto@macro{\appendix}{%   \addtocontents{toc}{\protect\renewcommand*{\protect\@chapapp}{\protect\appendixname}}% } \makeatother  \titlecontents{section}[5.3em] {\color{cyan}\large\itshape\addvspace{3mm}} {\contentslabel{0em}\hspace{3.2em}} {} {\hspace{3em}\contentspage} [\addvspace{0mm}]  %%% Chapter Header ---------------------------------------------------------- \makeatletter \titleformat{\chapter}   {\normalfont\sffamily\Huge\scshape}   {}{0pt}   {\thispagestyle{empty}    % Remove page number on new chapters   %\setcounter{definition}{0}   \begin{tikzpicture}[remember picture,overlay]     \node[yshift=-3cm] at (current page.north west)       {\begin{tikzpicture}[remember picture, overlay]         \draw[fill=cyan] (0,-1) rectangle           (25cm,3cm);         \draw[fill=cyan] (0,-24) rectangle           (25cm,-25cm);         \ifttl@label% &lt;---------------------- Added condition on \ifttl@label           \node[anchor=west,xshift=.21\paperwidth,yshift=-.01\paperheight,rectangle]               {\color{white}\LARGE \MakeUppercase{\@chapapp} \Huge\thechapter};         \fi% &lt;------------------------------- end condition on \ifttl@label        \end{tikzpicture}       };    \end{tikzpicture}\endgraf   \vskip-.7cm   \color{cyan}\Huge\raggedright\leftskip-1cm    \noindent\MakeUppercase{#1}\endgraf   } \makeatother  \titlespacing*{\chapter}{0pt}{10pt}{0pt}{}  \titleformat{\section}{\color{cyan}\itshape\LARGE}{\llap{\thesection} #1}{1em}{}[\titleline{\color{cyan}\titlerule[1pt]}] \titleformat{\subsection}{\color{cyan}\normalfont\large}{\; #1}{1em}{}  \begin{document}   \tableofcontents   \chapter{Turing Machines}     \section{The Turing Machine}     \section{Variations of the Turing Machine}       %\subsection{Non-deterministic Turing Machines}     \section{The Church-Turing Thesis}   \chapter{Decidability}     \section{The Universal Turing Machine}    \appendix   \chapter{ON COMPUTABLE NUMBERS, WITH AN APPLICATION TO THE ENTSCHEIDUNGSPROBLEM} \end{document} Allow given application to run as administrator, without prompting user for credentials I frequently cross this issue, and always have to google for an answer. Does anyone have a permanent fix for BADSIG errors from apt-get?     W: GPG error:    lucid   Release: The following signatures were   invalid: BADSIG 54422A4B98AB5139   Oracle Corporation (VirtualBox archive   signing key) As a result, I am able to be understanding and accommodating to the people that I work with, including clients, which I believe is an important asset for a professional to have.   I wanted to make sure it is absolutely known that I mean to acknowledge clients as a class of people I can work with. Is this the right way to include it with commas? How can I make sure that my on-screen colors are consistent? I designed a mockup on my LG monitor. When I view the same design on another monitor that I own, the colors seem drastically different. I reset both of the monitors to their factory defaults, but still the issue persists. I'm wondering if some of my clients might dislike my designs because of this. What should I do?  How can I make sure that what I design is what my client sees?  Even worse still, I've realized that my laptop is showing different colors too. I hava Lenovo g580 laptop. I want to change disk to SSD drive, but I'm not sure if my laptop support SATA 3 (6Gbps).  I write following output:  dmesg | grep SATA [    3.190743] ahci 0000:00:1f.2: AHCI 0001.0300 32 slots 6 ports 6 Gbps 0x5 impl SATA mode [    3.199450] ata1: SATA max UDMA/133 abar m2048@0xd3617000 port 0xd3617100 irq 43 [    3.199455] ata3: SATA max UDMA/133 abar m2048@0xd3617000 port 0xd3617200 irq 43 [    3.518365] ata1: SATA link up 3.0 Gbps (SStatus 123 SControl 300) [    3.857921] ata3: SATA link up 1.5 Gbps (SStatus 113 SControl 300)   Is that means I Can use SSD disk in SATAa 6Gbps mode? Using HTML5/Canvas/JavaScript to take in-browser screenshots Can a jar that has had botulinum food in it be cleaned If a home canned jar has botulinum in it can the jar be cleaned and saved? If so what is the correct way to clean the jar? Discount for Roles I am using Drupal Commerce and I would like to know if it is possible to have discounted prices for registered users and much better for roles.  For example, the nomral price is $100 but when a user logs in that belongs to Role1 for example, he will see different prices with the percentage discount that I will have assigned to that role.  How can I accomplish such a feature? the teacher gave me this problem to solve. So i hope you can help me with it. Here is the problem: John poured 150ml ob boiling water in 2 cups made out of polystyrene. Cups are open on the top so heat goes out there. Temperature of the outside air is 20°C. In 1 cup he add 5 mm thick layer of hot oil. He measured starting temperatures and then he measured temperatures again after 10 min. He measured the mass of water and water+oil before and after 10 min.  Here are the resaults:   -cup 1 (only water)  starting temp: 96°C  finnish temp: 64°C  starting mass: 149.4g  finnish mass: 142.42g  -cup 2(with oil on top)  starting temp.: 96°C  finnish temp.:91°C  starting mass: 164.72g  finnish mass: 164.70g  The surface of the both cups is 0.003 m2  Questions: 1.Why did the water covered with oil cooled down slower that the one with no oil? Show with calculations. 2.Why did John use oil with temperature around 100°C and not oil with room temperature? How would the resaults change if he used room temeprature oil? 3. Which other physical values, which are relevant, can you calculate from this informations? Basiccaly i need the physics behing why does this happen.