How to calculate Hamiltonian when Lagrangian has higher order derivatives?
How to obtain Hamiltonian formalism and phase space for Lagrangian with second-derivatives?
Prove universal morphism is unique up to unique isomorphism. I'm following along Wikipedia page() on universal property, and this seems it should be trivial, but I couldn't finish the proof.  The definition I am working with:   The problem   So I want to prove that if $(A, u)$ and $(A', u')$ are universal morphism from $X$ to $F$, then there exists unique isomorphism $k: A \rightarrow A'$ such that $u' = F(k) \circ u$.   My attempt:   For any $B \in C$ and $f: X \rightarrow F(B)$, there exists unique $h: A \rightarrow B$ such that $f = h \circ u$.  Letting $B = A'$ and $f = u'$, we see that there exists unique $k: A \rightarrow A'$ such that $Fk \circ u = u'$.  This shows existence and uniqueness. We only need that $k$ is isomorphism, but this is the part I cannot prove.   Similarly, there exists unique $k': A' \rightarrow A$ such that $Fk' \circ u' = u$.  Plugging into each equation, we get  $$Fk \circ Fk' \circ u' = F(k \circ k') \circ u' = u'$$  $$Fk' \circ Fk \circ u = F(k' \circ k) \circ u = u$$  but this of course doesn't imply $F(k \circ k') = id$, so it seems useless.   Thanks for your help! what could cause a script to fail to find python when it has `#!/usr/bin/env python` in the first line? Trying to get casperjs running on Ubuntu 12.04. After installing it when I run I get:   09:20 $ ll /usr/local/bin/casperjs lrwxrwxrwx 1 root root 26 Nov  6 16:49 /usr/local/bin/casperjs -&gt; /opt/casperjs/bin/casperjs  09:20 $ /usr/bin/env python --version Python 2.7.3  09:20 $ cat /opt/casperjs/bin/casperjs | head -4  #!/usr/bin/env python  import os import sys  09:20 $ casperjs : No such file or directory  09: 22 $ python Python 2.7.3 (default, Sep 26 2013, 20:03:06)  [GCC 4.6.3] on linux2   So Python is present and runnable, casperjs is pointing to the right place and it is a python script. But when I run it I get "No such file".   I can fix it by changing the first line of the casperjs python file from:    #!/usr/bin/env python   to:   #!/usr/bin/python   Result:   $ casperjs --version 1.1.0-DEV   I managed to fix it, but I'm wondering why it didn't work with #!/usr/bin/env python, since that seems to be a normal interpreter line. Do I have something configured wrong?   Here are the steps to get casperjs:   $ git clone git://github.com/n1k0/casperjs.git $ cd casperjs $ ln -sf `pwd`/bin/casperjs /usr/local/bin/casperjs $ casperjs : No such file or directory Can the Help action be used to give advantage to a specific ally's attack (rather than just the next ally who attacks the target)? The Impossible puzzle ("Now I know your product") I am trying to figure out this impossible problem by Martin Gardener and still havent found a suitable solution or link to it     Two mathematicians S and P are discussing two   unknown integers, both greater than 1. S knows only the sum of the   numbers, whereas P knows only their product.:      S: "I see no way you can determine my sum."    P (after a suitable delay): "That didn't help me. I still don't know your sum." S (after   another delay): "Now I know your product." What are the two numbers?   I have seen the answers on the xkcd wiki link for this at  say 13 and 16 and it checks out mathematically but then there are more ambiguous responses and most of the attempts seem brute force rather than logic.  My question is two fold   How to solve this problem logically? Are there more answers then 13,16 or this answer is the only one even as the upper bound k(upper bound on answer) shifts higher I recently found a formula to help find a sequence given a generating function:  $(\frac{1}{1-x})^k=\sum_{n=0}^{\infty}\binom{n+k-1}{n}x^n, k \in \mathbb{N}$  I just wondered if there was a known proof for this, or if anyone knows of a way to prove it. What is the white smoke-like fog called? I'm subscribed to the Ubuntu Security Notes, so everytime that I receive an e-mail, a new update is avalable.  But in the last update, they wrote something in the end that I coudn't undertand much, so can anyone please tell me what does that means?  ATTENTION: Due to an unavoidable ABI change the kernel updates have been given a new version number, which requires you to recompile and reinstall all third party kernel modules you might have installed. If you use linux-restricted-modules, you have to update that package as well to get modules which work with the new kernel version. Unless you manually uninstalled the standard kernel metapackages (e.g. linux-generic, linux-server, linux-powerpc), a standard system upgrade will automatically perform this as well.   I tried to ask to my friend (who also uses Linux), but he don't know much about kernel. Can I convert flat bar to drop bars on a hybrid bike Ideals in the ring of Gaussian integers What are the proper ideals, prime ideals, maximal ideals of $\mathbb{Z}[i]$, the ring of Gaussian integers.    Check whether $(1+i)$ is prime or maximal ideal.     Can someone help me please. I have no idea how to crack this problem. Thanks for your time. mysql-server-5.7 package post-installation script subprocess returned error exit status 127 I have a website server on Ubuntu server 18.04 with apache and mysql. The database stopped for no apparent reason. I just did a restart of mysql server to solve the issue but after a simple update I got some errors on mysql-server package. The update was for other packages and not mysql-server package. Mysql still stops at least once a week and each time I have to restarts the mysql-server.  Can some one help me to understand the error please ?  Here the result of: dpkg --configure -a  Configure mysql-server-5.7 (5.7.23-0ubuntu0.18.04.1) ... /var/lib/dpkg/info/mysql-server-5.7.postinst: ligne 191: /usr/share/mysql-common/configure-symlinks: no file or folder  dpkg: erreur de traitement du paquet mysql-server-5.7 (--configure) : installed mysql-server-5.7 package post-installation script subprocess returned error exit status 127 How are multiple volunteers handled during the Reaping? While skimming through the first few chapters of The Hunger Games to answer another question, I noticed this sentence when Katniss is describing the rules surrounding volunteers after Prim's name is called (at the beginning of the second chapter):     In some districts, in which winning the reaping is such a great honor, people are eager to risk their lives, the volunteering is complicated.   The only reason I can see for volunteering being complicated is if there's more than one volunteer for a particular year, and I don't recall this ever being addressed in the books.  Is there anything in the books that I've missed or any reliable, external sources (interviews or other supplementary material by the author) that explain exactly how this situation is resolved? Standard library sort and user defined types Quantum Entanglement - What's the big deal? Bearing in mind I am a layman - with no background in physics - please could someone explain what the "big deal" is with quantum entanglement?  I used to think I understood it - that 2 particles, say a light-year apart spatially, could affect each other physically, instantly.  Here I would understand the "big deal".  On further reading I've come to understand (maybe incorrectly) that the spatially separated particles may not affect each other, but in knowing one's properties you can infer the other's.  If that it the case, I don't see what the big deal is... 2 things have some properties set in correlation to each other at the point of entanglement, they are separated, measured, and found to have these properties...?  What am I missing? Is it that the particles properties are in an "un-set" state, and only when measured do they get set?  (i.e. the wave-function collapses).  If this is true - why do we think this instead of the more intuitive thought that the properties were set at an earlier time? Are there fields and ordered fields of every infinite cardinality? On the top of my head, I cannot think of any fields of cardinality more than that of the reals. (It is known that the process of algebraic closure does not increase the cardinality of an infinite field.)  What is the simplest way to give an example of a field (and an ordered field) of a specific cardinality $\alpha$?  I see there is the "Field" of surreal numbers, but it is a proper class rather than a set (and hence do not have a cardinality as such). However, there seems to be some modified construction which gives proper fields with the cardinality of some strongly inaccessible cardinal. What is the use of multiple main methods? What translation occurs when writing to a file that was opened in text mode that does not occur in binary mode? Specifically in MS Visual C.  unsigned char buffer[256]; for (int i = 0; i &lt; 256; i++) buffer[i]=i; int size  = 1; int count = 256;   Binary mode:  FILE *fp_binary = fopen(filename, "wb"); fwrite(buffer, size, count, fp_binary);   Versus text mode:  FILE *fp_text = fopen(filename, "wt"); fwrite(buffer, size, count, fp_text); Vim inoremap for specific filetypes How do I get a slanted \sum symbol? I want to use a slanted \sum symbol denoting a different meaning from summation. How do I get a slanted \sum symbol? Migrated questions don't have a user action label in the question list Finite Sum of Power?