Say we construct the Lagrangian for a system and minimise the action, what happens if this is not unique? In other words the action is minimised by two distinct (not infinitesimally separated) paths. Is there something else that governs the evolution of the system, or is there always a unique solution to the Euler-Lagrange equations?
A few months ago I was telling high school students about Fermat's principle.   You can use it to show that light reflects off a surface at equal angles. To set it up, you put in boundary conditions, like "the light starts at A and ends at B". But these conditions by themselves are insufficient to determine what the path is, because there's an extra irrelevant stationary time path, which is the light going directly from A to B without ever bouncing off the surface. We get rid of this by adding in another boundary condition, i.e. that we only care about paths that actually do bounce. Then the solution is unique.  Of course the second I finished saying this one of the students asked "what if you're inside an elliptical mirror, and A and B are the two foci?" In this case, you can impose the condition "we only care about paths that hit the mirror", but this doesn't nail down the path at all because any path that consists of a straight line from A to the mirror, followed by a straight line to B, will take equal time! So in this case the principle tells us nothing at all.  The fact that we can get no information whatsoever from an action principle feels disturbing. I thought the standard model was based on one of those!  My questions are   Is this anything more than a mathematical curiosity? Does this come up as a problem/obstacle in higher physics? Is there a nicer, mathematically natural way to state the "only count bouncing paths" condition? Also, is there a "nice" condition that specifies a path in the ellipse case? What should I have told that student?
Installing PHP 5.6 on Xenial (16.04) I need to use php5 (5.6) because I'm working on a Drupal 7 project which doesn't support php7. The problem is I recently updated to ubuntu 16.04 Xenial which seems to .   I tried to install it from , however when It try to install it with apt I always get similar errors.  $ sudo apt-get install php5-cli Cependant les paquets suivants le remplacent :   php7.0-cli:i386 php7.0-cli`   Which means it's obsolete. How can I install PHP 5.6 on Xenial? Deleting (Erasing) all Data from Failed SanDisk MicroSD Card I've had two 32GB SanDisk MicroSD cards go bad recently.  One I was able to temporarily revive using the kind advice offered in this thread:   When I insert the other into a reader, it registers as 30.6MB (that's MB, not GB).  I've tried all sorts of software and other techniques to revive this card and retrieve the important data off of it (including FindAndMount and TestDisk).  Nothing has worked.  So I'm thinking of returning the card for a refund.  But first, I would like to make it unreadable by someone more skilled than myself.  Is there any way to do this, given that I cannot even format the card or get it to be recognized as more than 30.6MB?  I am looking for a method that does not entail the kind of obvious physical destruction that would jeopardize obtaining a refund from the retailer.  If Costco (where the card was purchased) accepts physically destroyed cards for return, then physical destruction is acceptable (but please have knowledge that they do).  Given two failures, I don't really want to send the card to SanDisk for replacement, because I no longer trust the reliability of their products. Fitting linear model / ANOVA by group Quote.class doesn't return the Quote type, but Account.class does. Why? I'm trying a library that acts like an object factory. It works with serialization, so it is possible to specify formula fields without DML, which is pretty handy with tests.  My issue is that I'm trying to write a class that requires the Quote type. The apex compiler throws me the following error when using Quote.class:     Invalid field class for SObject Quote   But Opportunity.class, Account.class and even QuoteLineItem.class work just fine. Any idea why? Is it not possible to get the quote type? Or is there another way of doing it? Will ArcGIS for Desktop run on Windows 10? Will ArcGIS 10 run on the new windows 10 technical preview?  I am setting up an interface on my Mac and there is the free option of downloading the Windows 10 technical preview, some sort of beta version, I'm assuming.   Just wanted to see if 10.1 will run ok on Windows 10 (ignoring the Mac aspect), as this is all I am using my windows interface for? What determines how much experience I get from a hearthstone match in play mode? Sometimes I win handily and get a little experience, sometimes I squeak out a victory and get a lot.  How is experience earned? If $\gcd(a,b)=1$, then there exists integers $x$ and $y$ such that $xa + yb = 1$ Did not find this from this website...  If $$ \gcd(a,b)=1,$$ then there exists integers $x$ and $y$ such that $$xa+yb=1.$$  Now, the tip is to use particular corollary, that states:  The class $[m]_{n}$ generates $\mathbb{Z}/n\mathbb{Z}\Leftrightarrow \gcd(m,n)=1.$  I am totally lost with the corollary.  Let's assume that $\gcd(m,n)=1$. Then, $[m]_{n}$ generates $\mathbb{Z}/n\mathbb{Z}$.  OK! Then what?  There is also follow up, where I have to prove the converse. I am familiar with Bezout's lemma. Suppose $x&gt;0$ and we have function $f(x)=\frac{\sin x}{x}$ how can we show $$\forall n \in \mathbb{N} :|f^{(n)}(x)|\leq \frac{1}{n+1}$$ I need a hint to show this property .Thanks in advance . I tried for $n=1 ,2$  by finding maximum of $|f'| ,|f''|$ but I get stuck to show for $n$ A very recent post asking people to go "on strike" was just deleted Why does the flash freeze a picture? Why does a flash freeze a picture even if I use a shutter speed like 1/10 sec?  I don't understand it, because the light will hit the sensor the whole time, not just when the flash is fired. How do I set up Ruby on Rails? I am interested in setting up a Ruby on Rails environment on my 10.10 laptop.   Is there anything special I need to keep in mind for installation (i.e. that could be new or unique to 10.10)?   Are there any suggested walkthroughs that I could follow along with?  Thanks in advance. What's interesting in latus rectum? I'm a maths teacher in Italian secondary school and I've been spending some time trying to construct "meaningful" problems about conic sections. I particularly like problems which focus on practical applications of the curves (in engineering, physics, finance, etc.), on their geometric properties, or both.  I know there's this really easy formula to find the latus rectum of a conic from its key parameters. It's quite nice and straightforward to derive it, but...  ... What's its use? Why should anyone be interested in knowing the latus rectum of a curve? Is there any physical phenomenon, any architectural trick, or anything else where its position and/or length plays a key role?   I know just the relation between angular momentum, standard gravitational parameter and latus rectum in an astronomical orbit ($l=\frac{h^2}{GM}$, where $l$ is half the latus rectum, and $h$ is the ratio between the angular momentum associated to the orbiting body and its mass). It's cool, but a bit tricky, and probably too far-fetched for secondary school students... Is there anything simpler? Or just anything else, at any difficulty level?  Many thanks! Why am I getting "line 1: $' :\r': command not found"? I use Cygwin on my laptop (DOS). I have a collection of scripts from my colleagues, and my own. I am not an IT person, not knowledgeable in Unix. I am following my colleagues' syntax and able to manage a few simple things. The scripts worked well on my old laptop. I just changed laptop and installed Cygwin. When I run my scripts, they do not work. Here is one example of the error message I get: line 1: $':\r': command not found line 5: syntax error near unexpected token `$'\r'' line 5: `fi  Here are the top 5 lines of my script : iter=1 if [ -f iter.txt ]    then rm ./iter.txt  fi    Can someone please explain how I can get around this problem? Oneboxing a onebox from a different chat server generates a weird URL So if you try to onebox a onebox from a different chat server, . Basically, it generates this URL:    Now, obviously that isn't right. I've also tested this , and I'm assuming it's broken on chat.SO too. Can this be fixed? Show that $\textbf{ZF}$ shows $\phi$ is $\Delta_1$ Find joint density function of X and X+Y (exponential distribution) I tried finding the following density function:     Let $X$ and $Y$ be independent random variables, each having the exponential distribution with parameter $\lambda$. Find the joint density function of $X$ and $Z=X+Y$.   $\begin{aligned}f_{X,X+Y}=&amp;\frac{\partial}{\partial x\partial z}\mathbb P (X\leq x,Y\leq z-x)\\=&amp;\int_{-\infty}^x\int_{-\infty}^{z-x}f(u,v)\,\mathrm dv=\frac{\partial}{\partial z}\left[\frac{\partial}{\partial x}\int_0^x\int_0^{z-x}\lambda^2e^{-\lambda u-\lambda v}\,\mathrm dv\,\mathrm d u\right]\\ =&amp;\frac{\partial}{\partial z}\int_0^{z-x}\lambda^2e^{-\lambda x-\lambda v}\,\mathrm dv=\frac{\partial}{\partial z}\int_x^{z}\lambda^2e^{-\lambda x-\lambda(v-x)}\,\mathrm dv\\ =&amp;\lambda^2e^{-\lambda x-\lambda(z-x)}=\lambda^2e^{-\lambda z} \end{aligned}$  But I don't think this is correct, because $\mathbb P(X\leq x,X+Y\leq z)$ isn't equal to $\mathbb P(X\leq x,Y\leq z-x)$. But I can't think of anything else.  Could someone help me a little bit with this exercise?  I'm going to try the following:  $\mathbb P(X\leq x,Z\leq z)=\int_{-\infty}^x\int_{-\infty}^zf_X(u)f_Z(v)\,\mathrm du\,\mathrm dv$,  where I use the convolution formula. If I made no mistake, this gives  $f_{X+Y}(z)=z\lambda^2e^{-\lambda z}$, if $z&gt;0$.  Then calculating the joint density function:  $\begin{aligned}f_{X,X+Y}(x,z)=\frac{\partial}{\partial z}\left[\int_{-\infty}^z\lambda^3ve^{-\lambda x-\lambda v}\,\mathrm dv\right]. \end{aligned}$  Oh wait, this is also incorrect, because $X$ and $X+Y$ are dependent.  I really don't know how to proceed; could someone help?  I've also looked at this , but they mention the Jacobian, and my teacher has chosen to skip the paragraph about the Jacobian, so it wouldn't make sense if I had to use that. At what port did Lorance Effler born about 1787 in Germany come to the US? How to rename objects with python? Explanation of finite population correction factor? I understand that when sampling from a finite population and our sample size is more than 5% of the population, we need to make a correction on the sample's mean and standard error using this formula:  $\hspace{10mm} FPC=\sqrt{\frac{N-n}{N-1}}$  Where $N$ is the population size and $n$ is the sample size.  I have 3 questions about this formula:   Why is the threshold set at 5%? How was the formula derived? Are there other online resources that comprehensively explain this formula besides  paper? Let $f: \Bbb R \to \Bbb R$ be a differentiable function such that $\sup_{x \in \Bbb R}|f'(x)| \lt \infty$. Then (UGC CSIR-2015, DECEMEMBER, MATHEMATICAL SCIENCES)        $f$ maps a bounded sequence to a bounded sequence.   $f$ maps a Cauchy sequence to a Cauchy sequence.   $f$ maps a convergent sequence to a convergent sequence.   $f$ is uniformly continuous.      I choose all of the options as possible answers because the condition $\sup_{x\in \Bbb R}|f'(x)| \lt \infty$ forces $f$ to be uniformly continuous.(Because $f$ becomes Lipschitz and Lipschitz condition implies uniform continuity)  i.e. $\frac {|f(x)-f(y)|}{|x-y|} \le \sup_{x\in \Bbb R}|f'(x)|$ $ \forall x,y$.  Hence all other options are bound to be true.  Am I correct?