Goldstein's Classical Mechanics proposes two ways to derive the Euler-Lagrange (E-L) equations. One is by the D'Alembert's Principle of virtual work and the second is by Hamilton's Principle of Least Action. They are both motivated quite differently. The D'Alembert's principle is one that tries getting rid of the constraint forces by considering 'virtual displacements' -- for whom the work done by constraint forces is zero. He then procedes to change coordinates to a linearly independent set using the holonomic equations of constraint, which leads us to the E-L equations. Hamilton's principle is motivated quite differently. It looks at all possible paths in configuration space and picks the one in which the action is extremised. This, too, leads to the E-L equations. I wanted to know how/if the two methods are equivalent. As in, how does using virtual displacements and changing variables equate to extremizing a functional of a Lagrangian?
Why do we get the same differential equations from both principles? Surely there is a fundamental connection between them? When written out, the two seem to have nothing in common.   $$\sum _i ( \mathbf F _i - \dot{\mathbf p}_i) \cdot \delta \mathbf r _i = 0$$  $$S[q(t)] = \int ^{t_2} _{t_1} \mathcal L (q,\dot{q},t)dt$$  After playing with d'Alembert's principle  we find that we can rewrite the whole thing as $$\sum _i \left[ \frac{d}{dt} \left( \frac{\partial T}{\partial \dot{q}_i} \right) - \frac{\partial T}{\partial q_i}-Q_i\right] \delta q_i$$  This can further be rewritten under certain conditions so we get the exact form of the E-L equation.   It seems to me that both ways of arriving at the result are fundamentally different.   A function must obey the E-L equations in order to minimize the action over a path, but when we look at the virtual work, it appears that they come from the fact that (quoting Goldstein) "particles in the system will be in equilibrium under a force equal to the actual force plus a 'reversed effective force'."  I think I understand the principle of stationary action, I can see how it leads to the E-L equations, but d'Alembert's Principle seems so arbitrary, I can't see any motivation for it.
I'm looking for the best / friendliest way to reboot my rooted android 4.1.1 device. I found these solutions so far:   su -c "reboot" - Causes problems with wifi after restart on my device... I read that this can cause a currupt file system in some cases... Doesn't work via ssh for me. toolbox reboot - Same concerns here, but works via ssh. reboot -d 8 -f - causes problems after restart e.g. some apps wont start busybox killall system_server - "Hot Reboot" (not really a system restart) start|stop - Starts/stops Android Runtime (not really a system restart)   What is the device doing exactly when I'm pushing the (hardware) power button? I guess that's the most friendly way. I saw a lot of times claims that they have to be exhaustive (the examples in such books were always set in such way, that they were indeed), on the other hand I also saw a lot of times books stating they should be exclusive (for example $\mathrm{H}_{0}$ as $\mu_1=\mu_2$ and $\mathrm{H}_{1}$ as $\mu_1&gt;\mu_2$) without clarifying the exhaustive issue. Only before typing in this question I found somewhat stronger statement on the  -- "The alternative need not be the logical negation of the null hypothesis".  Could someone more experienced explain which is true, and I would be grateful for shedding some light on the (historical?) reasons for such difference (the books were written by statisticians after all, i.e. scientists, not philosophers). Intersection and union of simply connected domain I have some basics concept that, in $\mathbb{C}$, a "simply connected domain $D$" is a region in $\mathbb{C}$ with no holes. I am not sure whether it has various formal definition , but the one used in my complex analysis class looks very complicated. I barely understand what it means :  $\textbf{Definition}$ Let $\gamma_0, \gamma_1$ be two continuous closed path in a domain $D \subseteq \mathbb{C}$ parameterized by $I = [0,1]$;that is, $\gamma_i(0) = \gamma_i(1)$ for all $i = 0, 1.$ We say $\gamma_0, \gamma_1$ are $\textbf{homotopic as closed paths on D}$ if there exists function $\delta : I \times I \rightarrow D$ such that   $1) \ \delta(t,0) = \gamma_0(t) \ \forall t \in I \ \ $    $2) \ \delta(t,1) = \gamma_1(t) \ \forall t \in I \ \ $     $3) \ \delta(0,u) = \delta(1,u) \ \forall u \in I.$  A continuous closed path is $\textbf{homotopic to a point}$ if it is homotopic to a constant path(as a closed path).  $\textbf{Definition}$ A region $D \subseteq \mathbb{C}$ is called $\textbf{simply conected}$ if every continuous closed path in $D$ is homotopic to a point in $D$.  $\textbf{Problem :}$ 1) Let $D_1, D_2$ be simply connected plane domains whose intersection is nonempty and connected. Prove that their intersection and union are both simply connected.  2) Let $P, Q$ be smooth functions on a domain $D \subseteq \mathbb{C}$, Find necessary and sufficient condition for the form $P dz + Q d\bar{z}$ to be closed. Why not perma-ban instead of suspend an account for 275 years? I know there are a lot of questions about very long suspension periods, but I came across a case that is quite perplexing. This user had many offenses to earn his long ban, despite being a very prolific and high-rep user with many valuable posts, but the suspension period assigned strains the definition of "suspension."     This account is temporarily suspended network-wide. The suspension   period ends on Mar 18 '(22)92 at 16:28.   I can see why you might want to suspend an account for several years, maybe even a decade, if you don't believe behavior will change. However, what is the point of "suspending" an account for 275 years? Is there a reason why you can't just make it a permanent ban instead of pretending it is a suspension?    On a tangential note, isn't the "temporary" in "temporary suspension" redundant?  Suspend: temporarily prevent from continuing or being in force or effect. Avoiding jumping frames in beamer Last upgrade crashes network manager (no internet connection, no applet) While upgrading today, something caused my wireless network manager to disappear. Not even connecting via wired connection.  This error is being shown by files:  **(nm-applet:2716): WARNING **: Could not initialize NMClient /org/freedesktop/NetworkManager: The name org.freedesktop.NetworkManager was not provided by any .service files   (nm-applet:2716): nm-applet-WARNING **: Error connecting to ModemManager: Error calling StartServiceByName for org.freedesktop.ModemManager1: GDBus.Error:org.freedesktop.DBus.Error.Spawn.ExecFailed: Cannot launch daemon, file not found or permissions invalid   (nm-applet:2716): nm-applet-WARNING **: Could not find ShellVersion property on org.gnome.Shell after 5 tries   (nm-applet:2716): nm-applet-WARNING **: Failed to register as an agent: (2) The name org.freedesktop.NetworkManager was not provided by any .service files    Is there a patch I can download through windows and send through to my ubuntu, or is it a case of having to reload everything? I tried running two versions back but still no internet access. Module 'pdftexcmds' not found after update Yesterday I updated my distribution (tlmgr update --all) and now I cannot use TikZ with lualatex anymore.  Minimal working example:  \documentclass{article} \usepackage{tikz} \begin{document} a \end{document}   Compilation output:  &gt; lualatex a This is LuaTeX, Version 1.10.0 (TeX Live 2019)                                                                                                                                                                         restricted system commands enabled.                                                                                                                                                                                  (./a.tex                                                                                                                                                                                                              LaTeX2e &lt;2019-10-01&gt; patch level 3                                                                                                                                                                                     luaotfload | main : initialization completed in 0.098 seconds                                                                                                                                                         (/opt/texlive/2019/texmf-dist/tex/latex/base/article.cls                                                                                                                                                              Document Class: article 2019/10/25 v1.4k Standard LaTeX document class                                                                                                                                                (/opt/texlive/2019/texmf-dist/tex/latex/base/size10.clo))                                                                                                                                                             (/opt/texlive/2019/texmf-dist/tex/latex/pgf/frontendlayer/tikz.sty                                                                                                                                                    (/opt/texlive/2019/texmf-dist/tex/latex/pgf/basiclayer/pgf.sty                                                                                                                                                        (/opt/texlive/2019/texmf-dist/tex/latex/pgf/utilities/pgfrcs.sty              ...  (/opt/texlive/2019/texmf-dist/tex/generic/pgf/frontendlayer/tikz/tikz.code.tex (/opt/texlive/2019/texmf-dist/tex/generic/pgf/libraries/pgflibraryplothandlers. code.tex) (/opt/texlive/2019/texmf-dist/tex/generic/pgf/modules/pgfmodulematrix.code.tex)  (/opt/texlive/2019/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikz librarytopaths.code.tex))) No file a.aux. ABD: EveryShipout initializing macros (/opt/texlive/2019/texmf-dist/tex/context/base/mkii/supp-pdf.mkii [Loading MPS to PDF converter (version 2006.09.02).] ) (/opt/texlive/2019/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty[\direc tlua]:1: module 'pdftexcmds' not found:         no field package.preload['pdftexcmds']         [kpse lua searcher] file not found: 'pdftexcmds'         [kpse C searcher] file not found: 'pdftexcmds' stack traceback:         [C]: in function 'require'         [\directlua]:1: in main chunk. l.165       \directlua{require("pdftexcmds")}  ?    It seems that a Lua package is missing.  Is this a bug? Am I missing a package? REST API composite query: How do I use the result of a query subrequest in the referenceId of another subrequest? I am using Salesforce's REST API to (1) retrieve a list of object IDs according to a query, then (2) retrieve the records that correlate to those IDs.  I am familiar with the Salesforce composite documentation .    My request looks like this:  POST: /services/data/v48.0/composite  {     "allOrNone":false,     "compositeRequest": [ {         "method":"GET",         "url": "/services/data/v48.0/query?q=SELECT+Id+FROM+My_Entity+WHERE+(CreatedById+!=+'asdfghjkl')",         "referenceId": "myEntity"         },{         "method":"GET",         "url": "/services/data/v48.0/composite/sobjects/My_Entity?ids=@{myEntity.records[0].Id}",         "referenceId":"myObject"     }] }   My response:  {     "compositeResponse" : [ {         "body" : {             "totalSize" : 21,             "done" : true,             "records" : [ {                 "attributes" : {                     "type" : "My_Entity",                     "url" : "/services/data/v48.0/sobjects/My_Entity/a0D6g000xxxxxxx"                 },                 "Id" : "a0D6g000xxxxxxx"                 }, {*record2*}, {*etc*}             ]},             "httpHeaders" : { },             "httpStatusCode" : 200,             "referenceId" : "myEntity"         }, {         "body" : [ {         "errorCode" : "PROCESSING_HALTED",         "message" : "'myEntity.records' references an invalid Datatype. Only Strings and primitive data types are allowed to be referenced from previous operations."         } ],         "httpHeaders" : { },         "httpStatusCode" : 400,         "referenceId" : "myObject"     } ] }   The first subrequest (the query) executes without errors.  My problem is the second subrequest.  I understand from  that I can refer to the ID of a single item (e.g. myEntity.records[0].Id).  Question: How do I use all items from myEntity.records in the second subrequest? Euclidean and induction with prime numbers Why doesn't the last digit method work for divisibility by 4? I wish to explain to a grade 5 kid, the answer to the question - 'When can we know that the observing the last digit of a number will not be sufficient to determine its divisibility by a given N?'  For example - It doesn't work for 3, because we get all sorts of digits, i.e. from 0 to 9, at the units place. The same reason works for 7 and 9.   But what would be a convincing argument for 4 and 8? Even though the last digits form a pattern - 0, 2, 4, 6 and 8 - but we're still not sure about its divisibility. Loading OpenStreetMap data in QGIS? In QGIS 1.8 I used the openstreetmap plugin to request data from the server, or open a .osm XML file, which produced 3 layers (polygons, polylines and points) in QGIS.  This was great (give or take the 64bit bug which highly affects the freshly mapped area I work on), but how do I do that in QGIS 2.x?   I can't find any updated documentation about it.  In the vector menu, I've tried importing from osm servers, I get a .osm file, but then I'm stuck. There's a menu option for loading an XML file, but it produces a spatialite db, which apparently only has non-geometry tables when I try to add it as a spatialite layer. If I open the attribute table, I can see node ids, ways and so on, but I can't figure out how to actually make use of them.  I've also tried adding a vector layer from a .osm XML file. I get a gpsbabel window that shows nothing, then finally an invalid data source message in QGIS. After plain installation of Ubuntu 16.04 LTS gnome-software doesn't work. It has opened once with GUI, but installation of Unity Tweak Tool didn't finish.  When starting again, GUI doesn't come up. When opening from terminal, a warning appears, but the application GUI doesn't open.  $ gnome-software (gnome-software:7052): Gs-WARNING: failed to open plugin /usr/lib/gs-plugins-9/libgs_plugin_xdg_app_reviews.so: /usr/lib/gs-plugins-9/libgs_plugin_xdg_app_reviews.so: cannot open shared object file: No such file or directory   Does anybody have same experience, and eventually, a solution?    According to the recommendation:  $ dpkg-query -S /usr/lib/gs-plugins-9/libgs_plugin_xdg_app_reviews.so gnome-software: /usr/lib/gs-plugins-9/libgs_plugin_xdg_app_reviews.so   Proposed reinstall of gnome-software did not help.    As workaround, use Synaptic Package manager and command line. Sharing a Bluetooth Keyboard and Mouse sequentially between two Macs Find the least value of $ \sec^6 x +\csc^6 x + \sec^6 x\csc^6 x$ Find the least value of $$ \sec^6 x +\csc^6 x + \sec^6 x\csc^6 x$$ I tried AM greater than equal to GM But that's for finding maximum value.  This can probably be solved with calculus but I don't know for some reason I can't find the answer . Which is $80$. I'm trying to solve a question where I need to write $\frac {1}{1+z^2} $ as a power series centered at $z_0=1$  I'm not allowed to use taylor expansion. So my first thought was to rewrite the function in a form where I can apply the basic identity:  $$ \frac{1}{1-x} = \sum_{n=0}^\infty x^n $$  So let's rewrite the original function $f(z) = \frac {1}{1+z^2} $ and use above identity:  $$\frac {1}{1+z^2} = \frac {1}{1-(-z^2)} $$ But observe that I need to center it at $z_0=1$ so I rewrote it to:  $$\frac {1}{1-(-z^2)}=\frac {1}{2z-(-(z-1)^2)} $$  I can factor out $\frac {1}{2z}$ and use the basic identity to get : $$\frac {1}{2z}\frac {1}{1-\frac{(z-1)^2}{2z}} = \frac {1}{2z}\sum_{n=0}^\infty \left(\frac{-(z-1)^2}{2z}\right)^n =\frac {1}{2z}\sum_{n=0}^\infty \frac{(-1)^n(z-1)^{2n}}{(2z)^n} = \sum_{n=0}^\infty \frac{(-1)^n(z-1)^{2n}}{(2z)^{n+1}}$$  The problem is if I ask wolfram alpha: power series$ \frac{1}{1+z^2} $centered at z = 1 it will give me a different answer that doesn't match the sum I found.  Can anyone help me figure this one out? What am I doing wrong? I'm referring to the colored circles with numbers next to them on the top of the page, to the right of my reputation counter. What do they represent? Badge icons missing in Achievements of mobile view The badge icon followed by the site icon in the Achievements dialog is missing in the mobile view. But the same is available in the full site view.  Screenshots for reference:  Achievements - Full site view:        Achievements - Mobile view: Uses of the definite article (the) in generic noun phrases I was reading a paragraph about lions and I came up with a question about the definite article (the).  Let me tell you first what I know about it.     1->We use the before a singular noun (when we are sure about the noun.    And the listener and the speaker both know about it).      2->We use the before a plural noun (again when we are sure about the noun,   and we are not talking generally).   So, my question is why does the writer use the here as we are not talking about any specific lion or lions. We are talking generally.  The Paragraph:     Living in the grasslands, scrub, and open woodlands of sub-Saharan   Africa, the lion is the second largest cat in the world. It is dwarfed   slightly by the tiger, which is closely related and has a very similar   body type.    Can we re-write the paragraph something like this:     Living in grasslands, scrub, and open woodlands of sub-Saharan Africa,   lions are the second largest cat in the world. They are dwarfed   slightly by tigers, they are closely related and has a very similar   body type. Can I carry spare Lithium-Ion batteries with me on Lufthansa? I am travelling from the US to India on Lufthansa. I have five spare Lithium-ion batteries, each 850mAh.  Can I bring them back safely? Will wolves attack and kill ocelots in Minecraft? I know they kill sheep. I just don't want my cats being eaten.