We have recently covered the Lagrangian in our lectures, whereby it was shown that all equations of motion ($x(t)$) satisfying the Euler-Lagrange equation with Lagrangian $L=T-V$, where $T=\frac{1}{2}mv^2$ and $F=-\frac{\partial V}{\partial x}$, must also satisfy $F=m\ddot{x}$.  However, to me, it wasn't intuitive at all why this must have been the case: why, if we minimise (or maximise) some arbitrary quantity $L$, which didn't even have to be conserved, we for some reason get Newton's Second Law, which also implies that $L$ is in fact energy and has to be conserved.  Is there any intuitive way of understanding why ALL possible types of motion MUST minimise this arbitrary quantity $S$ and why, if we do minimise it, we get that $L$ is conserved and happens to represent the total energy of an object?
I'll be generous and say it might be reasonable to assume that nature would tend to minimize, or maybe even maximize, the integral over time of $T-V$.  Okay, fine.  You write down the action functional, require that it be a minimum (or maximum), and arrive at the Euler-Lagrange equations.  Great.  But now you want these Euler-Lagrange equations to not just be derivable from the Principle of Least Action, but you want it to be equivalent to the Principle of Least Action.  After thinking about it for awhile, you realize that this implies that the Principle of Least Action isn't really the Principle of Least Action at all:  it's the "Principle of Stationary Action".  Maybe this is just me, but as generous as I may be, I will not grant you that it is "natural" to assume that nature tends to choose the path that is stationary point of the action functional.  Not to mention, it isn't even obvious that there is such a path, or if there is one, that it is unique.  But the problems don't stop there.  Even if you grant the "Principle of Stationary Action" as fundamentally and universally true, you realize that not all the equations of motions that you would like to have are derivable from this if you restrict yourself to a Lagrangian of the form $T-V$.  As far as I can tell, from here it's a matter of playing around until you get a Lagrangian that produces the equations of motion you want.  From my (perhaps naive point of view), there is nothing at all particularly natural (although I will admit, it is quite useful) about the formulation of classical mechanics this way.  Of course, this wouldn't be such a big deal if these classical ideas stayed with the classical physics, but these ideas are absolutely fundamental to how we think about things as modern as quantum field theory.  Could someone please convince me that there is something natural about the choice of the Lagrangian formulation of classical mechanics (I don't mean in comparison with the Hamiltonian formulation; I mean period), and in fact, that it is so natural that we would not even dare abandon these ideas?
Prove that $p \in \mathbb{R}[x]$ can be represented as a sum of squares of polynomials from $\mathbb{R}[x]$ My windows directory is huge I have always had problems with my windows directory's growing to a huge size.  It started with windows 95 then 98 then 2000 and xp.  Windows 7 however is absolutely huge.  The windows folder itself with out program files is 21.4 gigs...add the program files and other windows files and it is a whopping 32 gigs.  My question is why is it so large and what can I do about it.  At the rate this is going I will be out of space in now time. How to input open intervals I want to write open and half-open intervals using the following notation:  ]a, b[ ]a, b] ]-∞, b]   When writing them just like that in the LaTeX source, the spacing doesn't come out right. For example, for this equation:   \(X = ]-\frac{π}{2}, +\frac{π}{2}[\)   the minus sign is typeset as a binary operator. I can solve this by surrounding the whole value in braces, but then the spacing around the equal sign is still not correct. For other combinations of intervals and operators, different spacing inconsistencies arise.  What is the correct and easy  way to input intervals in order to avoid having to take care of this at all? I'm looking for a formula to calculate the sum of $(a+b+c+d+...)^n$ but with coefficients equal to 1. For example in $(a+b+c)^2$. I want the sum of $a^2 + b^2 + c^2 + ab + bc + ca$. And for $(a+b+c+d)^3$, I want the sum of $a^3 + b^3 + c^3 + a^2b + a^2c + b^2a + b^2c + c^2a + c^2b + abc$. Similarly, I want the sum of expansion of $(a+b+c+d+e+....)^n$ with coefficient equal to 1. I tried to find the pattern by expanding the $(a+b+c)^2$. And I found that its formula is $a^2 + b^2 + c^2$ + $\frac{((a+b+c)^2 - a^2-b^2-c^2)}{2}$. This works for the power of 2. But fails on other powers. Any help will be really appreciated. Very strange error suddenly appeared running Biber I am on a Mac with MacTex 2020 and ran into a strange problem with Biber. Working on a document all was fine, I did not change the bibliography (says my git) and only fixing typos (not even adding cites) and suddenly latexmk stopped working after a Biber run, while that looked like  INFO - This is Biber 2.14 INFO - Logfile is 'doc.blg' INFO - Reading 'doc.bcf' INFO - Found 31 citekeys in bib section 0 INFO - Processing section 0 INFO - Globbing data source 'literature.bib' INFO - Globbed data source 'literature.bib' to literature.bib INFO - Looking for bibtex format file 'literature.bib' for section 0 INFO - LaTeX decoding ... INFO - Found BibTeX data source 'literature.bib'   so harmless in my view. Then with --debug I additionally get in the end.  INFO - LaTeX decoding ... warning: overriding existing definition of macro "jun" warning: overriding existing definition of macro "jul" warning: overriding existing definition of macro "sep" warning: overriding existing definition of macro "jan" [...] warning: overriding existing definition of macro "nov" INFO - Found BibTeX data source 'literature.bib' Unicode::UCD: failed to find unicore/version in /var/folders/hm/gt63cn0j2r7bwbzw7c1hww1r0000gn/T/par-626572676d616e6e/cache-2f8a664c3c933a44d096bcc8d018a5e92bced897/inc/lib /var/folders/hm/gt63cn0j2r7bwbzw7c1hww1r0000gn/T/par-626572676d616e6e/cache-2f8a664c3c933a44d096bcc8d018a5e92bced897/inc CODE(0x7fd026aeb5a8) CODE(0x7fd026aebb90) at /var/folders/hm/gt63cn0j2r7bwbzw7c1hww1r0000gn/T/par-626572676d616e6e/cache-2f8a664c3c933a44d096bcc8d018a5e92bced897/inc/lib/Biber/Input/file/bibtex.pm line 1011.   It seems unrelated to the library itself since deleting a few entries (going through deleting one half, then the other), the problem persists. It does not happen for an empty literature, but as soon as a first entry is added (just author/journal/title/year/volume/pages - no utf8-only chars) this error message appears.  what does this last error message mean?  How can I solve this? Don't worry, I have written your name on the attendance sheet.   Or     Don't worry, I wrote your name on the attendance sheet. 80's/90's movie: man and woman driving through a red desert planet I was laying on my bed, reading a book when the sun shone through the windows on my left. I happened to look at the wall on my right and noticed this very strange effect. The shadow of my elbow, when near the pages of the book, joined up with the shadow of the book even though I wasn't physically touching it.   I have posted :The video seems to be the wrong way up, but you still get the idea of what is happening.  What is causing this? Some sort of optical illusion where the light gets bent?  Coincidentally, I have been wondering about a similar effect recently where if you focus your eye on a nearby object, say, your finger, objects behind it in the distance seem to get curved/distorted around the edge of your finger. It seems awfully related...  EDIT: I could see the bulge with my bare eyes to the same extent as in the video! The room was well light and the wall was indeed quite bright. How to reduce the spacing between responses in the alterqcm package? This question has a continuation here   The alterqcm package is no longer maintained by Alain Matthes. Two other developers maintain it. I wrote to them 15 days ago by email to report the error reported here:  I have not received a reply. I am trying to reduce the spacing between the lines of the QCM as indicated in  by @Werner and it does not work. In addition, by using the David Carlisle's code given in , I get an X instead of a square.  Full code of the screenshot: \documentclass[12pt]{article} %\usepackage[utf8]{inputenc} %\usepackage[upright]{fourier} \usepackage{etoolbox}% http://ctan.org/pkg/etoolbox \usepackage[english]{alterqcm} \def\square{X} % &lt;---------- David Carlisle comment \def\nogreekalph{??} % &lt;---- David Carlisle comment \parindent0pt \begin{document} % Original display \begin{alterqcm}[lq=8cm]  \AQquestion{Question}{%  {Proposition 1},  {Proposition 2},  {Proposition 3}} \end{alterqcm}  \bigskip  % Smashed display !!!!!! Dont work now  {\begin{alterqcm}[lq=8cm]  \AQquestion{Question}{%  {\smash{Proposition 1}},  {\smash{Proposition 2}},  {\smash{Proposition 3}}} \end{alterqcm}}  \bigskip  % Global smashed display !!!! Dont work now  \makeatletter \patchcmd{\aq@prop}{\vbox{#1}}{\vbox{\smash{#1}}}{}{}% patch \aq@prop \makeatother \begin{alterqcm}[lq=8cm]  \AQquestion{Question}{%  {Proposition 1},  {Proposition 2},  {Proposition 3}} \end{alterqcm}  \end{document}  Is there a solution? This question is here just to avoid you choosing the wrong Ubuntu version number.  So, this is not about:     nor about   nor   nor   nor   nor .   So in the answer below a simple guide as to what Ubuntu version is good for you, independent of future releases! Send request for "reason for down vote"? Each question in SO keeps a list of the users who voted it down. As we all know, the list is never shown to anyone. Only the down voter can see which questions he down voted.   Sometimes, down voters down vote rather hastily. Thats okay because no one is perfect (except chuck norris). But sometimes, the down voter might have a good reason to down vote but may not leave a reason behind. As a result, the poster is left wondering if his question is stupid/bad or okay. To remedy this situation, I suggest a feature below -   The poster should have a button "request down voters for reason" which can be pushed only once. The down voters in the down voter list get the notification and can choose to respond to it. I feel that this can serve as a polite reminder to those who down vote a question which would seem to be an okay (if not good) question to most people.   Example of question which I feel should have at least 0 votes -    If okay, then this feature can be combined with -    Do you agree to the proposal ?  As an aside, do you agree that the SO question above should not get negative points ? Find the last $2$ digits of $$\large7^{7^{7^{10217}}}$$   So far I have: $17\cdot 601=10217$ and $7=7 \pmod{10}$. Any help greatly appreciated. Custom Post Type Archives by Year & Month? How do you display the archives of a Custom Post Type by Year &amp; Month? How to format table with long column head entries? How do I calculate the probability density function for a joint beta + uniform distribution? The PDF of a Beta distribution is $$ f_X(x) = {{x^{a-1}(1-x)^{b-1}}\over {B(a,b)}}  $$ and CDF $$ F_X(x) =  I_x(a,b) $$ The PDF of a uniform distribution is $$ f_Y(y) =  {1 \over {b-a}} \space for \space  x \in [a,b] $$ and CDF $$ { x - a} \over {b - a} $$ I've know that if X and Y are two independent, continuous random variables, described by probability density functions above of $f_X(x)$ and $f_Y(x)$ then probability density function of $Z = XY$ is $$  f_Z(z) = \int_\infty^\infty f_X(x)f_Y({z\over x}){1\over abs(x)}dx  $$ but I'm struggling with the algebra to get  from this to the PDF and the CDF of the joint distribution. I'm using the uniform distribution between 0 and 1. I asked a question  but, rather stupidly, added extra information after it had been answered.  My additional question is this: I have a table with varying lengths of words. I want to encircle different groups of cells, including diagonals. If possible, it would be great if the rectangles don't overlap each other where possible.  I looked at the answer , which, whilst tedious, worked except for the overlap.    This is where I've got to so far:  \documentclass[fleqn,12pt,a4paper,landscape]{article}  \usepackage{tikz} \usepackage{tabularx}  \begin{document}  % Introduce a new counter for counting the nodes needed for circling \newcounter{nodecount} % Command for making a new node and naming it according to the nodecount     counter \newcommand\tabnode[1]{\addtocounter{nodecount}{1} \tikz \node     (\arabic{nodecount}) {#1};}  % Some options common to all the nodes and paths \tikzstyle{every picture}+=[remember picture,baseline] \tikzstyle{every node}+=[inner sep=0pt,anchor=base, minimum width=1.8cm,align=center,text depth=.25ex,outer sep=1.5pt] \tikzstyle{every path}+=[thick, rounded corners]  \begin{table}[ht]   \begin{minipage}[b]{1 \linewidth}\centering    \begin{tabular}{cccccc} % Turn the cells needed for defining the circling paths into nodes with our     custom command \tabnode{Long ago}&amp; \tabnode{A few days ago}&amp; \tabnode{Day before         yesterday}&amp; \tabnode{Yesterday} &amp;  &amp;  \\      &amp; &amp; &amp;  &amp; \tabnode{Today} &amp; \tabnode{Now}\\      \tabnode{Long time in the future}&amp; \tabnode{A few days on}&amp;     \tabnode{Day after tomorrow}&amp; \tabnode{Tomorrow} &amp;  &amp; \\      \end{tabular}    \end{minipage}   \begin{tikzpicture}[overlay] % Define the circle paths \draw [blue](1.north west) -- (1.north east) -| (7.north east) |- (7.south     west) |- (1.north west); \draw [red](8.south west) -| (2.north east) -| (2.south west) |- (8.south     west); \draw [yellow](9.south west) -| (3.north east) -| (3.south west) |- (9.south     west); \draw [green](8.south west) -| (3.north east) -| (2.south west) |- (8.south     west);   \end{tikzpicture}  \end{table} \end{document}  %%% Local Variables:  %%% mode: latex %%% TeX-master: t %%% End: etags and other tags The  tag is a synonym of , and has the description     A program to generate tags for a large project. These tags are usable by the Emacs text editor.   This is inconsistent: tags in that sense are always plural, i.e. etags, never etag.  evidently refers to another concept, .  Should  remain a synonym of , and be about HTTP Etags, and should Emacs etags be tagged +? Or should the synonym be unmade, with HTTP Etags on  and etags on ? Radius of Convergence of a given Power series What are the general rules regarding when can you refreeze food that was frozen and then thawed in the fridge? and what are the reasons behind them? (Hygiene? Food quality?). Let's assume for the sake of the argument that the items were in a non-frozen state only a few hours.  To be more specific, I'm interested in the application of these rules for  - Ginger garlic paste - Dough - Indian Curries - Green Chilly paste - Frozen green peas What can cause "Android OS" process to use high percentage of battery?