Do the same equations of motion imply the same Lagrangians? If two Lagrangian (densities) $\mathcal{L}$ give the same equations of motion, are they equivalent?
Proof that total derivative is the only function that can be added to Lagrangian without changing the EOM 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?
convert a html book to a pdf file? How can I convert a html book (consisting of a number of html files linked to each other) to a pdf file on Ubuntu?   Hope the resulting pdf file can have bookmarks according to the structures of the html book.  The html book can be downloaded from . Extract it, and go to help, and the directory manual is the html book. is there an easy way or photoshop plugin to distribute words in evenly spaced formation Show that $\lim _{r \to 0} \|T_rf−f\|_{L_p} =0.$ I'm not sure if there's a better place to ask this or not.   Why, at least here in America, is the standard electrical system set up for alternating current (AC) rather than direct current (DC)? I'm just completing a long-distance move and realizing how many DC adapters I have and how many of my devices require DC power:   All of my guitar pedals. My alarm clock. My external hard disks. My USB car HDD dock.   ... and the list goes on. If we're almost constantly having to use DC adapters to convert AC into DC, why isn't the standard DC, other than because people resist change? Output two sided, but only on recto page -- occasional images on verso I am preparing a thesis. In order to reduce the number of physical sheets of paper, I need to print certain figures/tables on the verso side. However the university regs state that, while figs/tables may appear on the verso, the body text must appear on the recto side.  I would like to therefore output the vast bulk of the work on the recto page, but with occasional verso pages. Ideally they would be numbered as a "subpage" of the preceeding recto page (e.g. 131, 131a, 132...) or similar.  The approach which I will tentatively adopt is to use a "twoside" flag, then have "empty" and non-numbered verso pages until a page with content is required.   I'm sure I'm not the first person to have had a similar issue, but am unable to imagine other ways of posing the question. Fermat: Last two digits of $7^{355}$ I am doing this problem mentioned above and I know the answer because I know Euler's Theorem that $$a^{\varphi{(m)}}\equiv{1}\pmod{m}.$$  I used 100 as my modulus and got that the last two digits of $7^{355}$ are $43$.  My question is this.  What prime should I pick to solve this problem using Fermat's theorem $$a^{p-1}\equiv{1}\pmod{p}$$  As a precursor problem, the author (Dudley; Elementary Number Theory) asked to find the last digit of the number above; so using a modulus $5$, $7^4\equiv{1}\pmod{5}$ by Fermat.  Since the answer is $7^{355}\equiv{3}\pmod{5}$, the last digit is either $3$ or $8$.  Since $7$ is odd, the answer will clearly be odd, so $3$ is it.  The book is working linearly, building from the previous chapters and Euler's Theorem isn't mentioned for another two chapters (I remembered Euler from a problem solving class in college).    So again, what prime would be best?  How should I procede?    EDIT: I've seen a bunch of problems on the site and none that I looked at did it using Fermat.  If there is one, that could be posted and I would be happy.  EDIT 2:  it seems that the problem of finding of finding the last two digits using is rather difficult because of the implication that we need something mod 100.  Is there a way around this , like adding a step or complexifying, in order to finish the problem with Fermat? So I decided to install my copy of Ableton (v9.1) on my computer now that I've upgraded to a machine that isn't ancient. HOWEVER Despite WineHQ reporting that Ableton is platinum-certified, I can't get the dern thing to open.  It crashes out just as the splash screen pops up.  Sometimes I see it, sometimes not.  It tells me that some catastrophic error has occurred, and it needs to close.  I've tried this on two different systems running two different versions of wine  (One runs the up-to-date public distro 1.6.2, while the machine I actually want to install it on is running the 1.7.19)  I don't know what to do.  I'd really just like to get this working, as my friends really like the look and feel of Ableton, and I really don't like LMMS (Linux-y free Audio Software).  Any idea what's going on here? I've found a couple of other people who seem to be having the same problem as me, but no dice on any solutions.  None provided, really.  Specs: Ubuntu 14.04 LTS  8 Gig ram  i7-2670QM CPU @ 2.20GHz × 8  Lots of free space... yadda yadda yadda   Error message: xgboost tayler expansion detail This is the objective function for Xgboost.  I have no idea where $g_{i}$ and $h_{i}$ came from  is some one explain how this two terms came form?  or direct me to the related tutorial page then I will search for it How do I disable the guest session? Java Enum definition I thought I understood Java generics pretty well, but then I came across the following in java.lang.Enum:  class Enum&lt;E extends Enum&lt;E&gt;&gt;   Could someone explain how to interpret this type parameter? Bonus points for providing other examples of where a similar type parameter could be used. What's a word for lack of confidence? I'm trying to find a word that describes a lack of confidence. Something that gives a sense of worthlessness maybe. I've dried looking up in Thesarus.com for similarities to lack of confidence and antonyms to self-worth, but with no luck. Anyone have some ideas? If $m,n\in N$ Prove that there is such a positive integer k, such that $(\sqrt{m}+\sqrt{m+1})^n=\sqrt{k}+\sqrt{k+1}$ If $m,n\in N$ Prove that there is such a positive integer k, such that $(\sqrt{m}+\sqrt{m+1})^n=\sqrt{k}+\sqrt{k+1}$ I attempted to solve this question using binomial coefficients, saying: $(\sqrt{m}+\sqrt{m+1})^n=\sqrt{k}+\sqrt{k+1}=\sum\limits_{a=0}^n {n\choose a}*\sqrt{m}^{n-a}*\sqrt{m+1}^a$ and from here I was thinking that I had to do something with $\sqrt{m}^{n-a}$ with $\sqrt{k}$ and do something else with $\sqrt{k+1}$ and $\sqrt{m+1}^a$. Unfortunately I couldn't think of what to do with this circumstance. I then thought of maybe taking $(\sqrt{k}+\sqrt{k+1})^2=(\sum\limits_{a=0}^n {n\choose a}*\sqrt{m}^{n-a}*\sqrt{m+1}^a)^2$, but this immediately over-complicated the question. Could you please explain to me how to solve this question and how to solve similar questions in the future? How should I add users non-interactively? I have a set of users I need to add to a new system as part of a BASH script that preps my server. How can I non-interactively add the user, create their home directory, set the group, and enter a default password?  For example, if I needed to add the following users, how would I go about that?  user: node password: spoon group: ultraMegaPermissions  user:serve password: spoot group: sortaNotMega  user:watcher password: spoogle group: notMega Since multiple inheritance is bad (it makes the source more complicated) C# does not provide such a pattern directly. But sometimes it would be helpful to have this ability. For instance I'm able to implement the missing multiple inheritance pattern using interfaces and three classes like that: public interface IFirst { void FirstMethod(); } public interface ISecond { void SecondMethod(); }  public class First:IFirst  {      public void FirstMethod() { Console.WriteLine(&quot;First&quot;); }  }  public class Second:ISecond  {      public void SecondMethod() { Console.WriteLine(&quot;Second&quot;); }  }  public class FirstAndSecond: IFirst, ISecond {     First first = new First();     Second second = new Second();     public void FirstMethod() { first.FirstMethod(); }     public void SecondMethod() { second.SecondMethod(); } }  Every time I add a method to one of the interfaces I need to change the class FirstAndSecond as well. Is there a way to inject multiple existing classes into one new class like it is possible in C++? Maybe there is a solution using some kind of code generation? Or it may look like this (imaginary c# syntax): public class FirstAndSecond: IFirst from First, ISecond from Second { }  So that there won't be a need to update the class FirstAndSecond when I modify one of the interfaces.  EDIT Maybe it would be better to consider a practical example: You have an existing class (e.g. a text based TCP client based on ITextTcpClient) which you do already use at different locations inside your project.  Now you feel the need to create a component of your class to be easy accessible for windows forms developers. As far as I know you currently have two ways to do this:  Write a new class that is inherited from components and implements the interface of the TextTcpClient class using an instance of the class itself as shown with FirstAndSecond.  Write a new class that inherits from TextTcpClient and somehow implements IComponent (haven't actually tried this yet).   In both cases you need to do work per method and not per class. Since you know that we will need all the methods of TextTcpClient and Component it would be the easiest solution to just combine those two into one class. To avoid conflicts this may be done by code generation where the result could be altered afterwards but typing this by hand is a pure pain in the ass. Relation between intensity of light and refractive index How do I install drivers for Realtek RTL8812BU? I've checked  for acute:  an acute illness is one that has quickly become severe and dangerous Compare:CHRONIC  And for chronic:  lasting for a long time Compare:ACUTE  Can a disease be both &quot;acute&quot; and &quot;chronic&quot; simultaneously? For example it could be the case where a certain illness quickly becomes serious and have long term consequences Are there any &quot;obtuse illnesses&quot; ? Types of books or Types of book? can anybody tell me which one of these is the correct one? I have been searching all over the internet but I have still got no answer.  Types of books Types of book Someone asked me this question and I don't think I gave him an adequate answer (I was trying to think of the extreme case - that of neutron stars) How does Loki survive his fall at the end of Thor? In one of the concluding scenes of the movie Thor, it's shown that Loki deliberately let go of the Scepter, and falls deep into the vast intergalactic sea sort of thing. How does he survive and return in The Avengers?