If we have a Lagrangian density $\mathcal{L}$ for a scalar field $\phi$ depending on $\phi$, $\partial _{\mu} \phi$, and $\partial _{\mu} \partial _{\nu} \phi$, what is the Hamiltonian? Additionally, what do Hamilton's equations become?
This is a special case of , which, I think, might have drawn little attention because it was too general. In this question, I would like to consider a specific case.  Take a classical system given by the action $$ S[x] = \int dt \left[ \frac{\alpha}{2} \ddot{x}^2 + \frac{\beta}{2} \dot{x}^2 + \frac{\gamma}{2} x^2 \right]. $$  This action is Lagrangian, but it is not what we are usually dealing with in physics, because the Lagrangian contains second time derivatives.  The equations of motion are:  $$ \alpha \ddddot{x} - \beta \ddot{x} + \gamma x = 0. $$  This can be solved as usual by employing Fourier transform:  $$ x(t) = a_1 e^{i \omega_1 t} + a_1^{*} e^{-i \omega_1 t} + a_2 e^{i \omega_2 t} + a_2^{*} e^{-i \omega_2 t}, $$  where $\omega_1^2$ and $\omega_2^2$ are the two roots of $\alpha \omega^4 + \beta \omega^2 + \gamma$, given by  $$ \omega_{1,2}^2 = \frac{- \beta \pm \sqrt{\beta^2 - 4 \alpha \gamma}}{2 \alpha} .$$  Now by definition the phase space is the space of solutions of equations of motion. In this case it is parametrized by two complex co-ordinates $a_{1,2}$, meaning that it is 4-dimensional.  I would like to know if there's a natural way to associate the symplectic structure (Poisson bracket) on this phase space. I.e.,  $$ \{a_1, a_1^{*}\} = ? $$ $$ \{a_1, a_2^{*}\} = ? $$ $$ \{a_2, a_1^{*}\} = ? $$ $$ \{a_2, a_2^{*}\} = ? $$ $$ \{a_1, a_2\} = ? $$ $$ \{a_1^{*}, a_2^{*}\} = ? $$
According to my text book:  The Lorentz Force F on a charged particle moving in an electromagnetic field is given by:  $$F= qE + q(v\times B)$$  then it states that "it is to be pointed out that only the electric force does work, while the no work is done by the magnetic force which is simply a deflecting force."  How can this by true won't the the magnetic force have a considerable magnitude, will it now impart kinetic energy to the charge? Any explanations will help a lot. Please keep the terminology simple, thanks. Visiting Russia as an Arab. Am I going to be hassled? Compute the sum of the series $\sum_{n=1}^{\infty} \frac{1}{n \cdot 2^n}$ What would be the sum of the following series?  $$\sum_{n=1}^{\infty} \frac{1}{n \cdot 2^n}$$  Thanks Unless I missed something key during Descent 1/2, how is it that the Borg are once again a collective? I want to create a animation like those "spacetime deformation" ones, but I am stuck on creating "spacetime".  (a mesh/plane).  I created a plane, gave it a soft body, collisions, and a passive rigidbody.    A cube with a rigidbody and collisions will land and deform the mesh, but it snaps right back.  How can I make it so the cube lands, and then leaves the plane deformed, and the cube just sits there?  I googled around and searched this site, nothing comes up.  An image of what I want (after it settles):    The ball depressed the mesh, creating an indentation. PCIe Bus error severity=Corrected I have a new HP Pavilion Gaming Notebook and a new installation of Ubuntu 16.04. When I press Ctrl + Alt + F1 I go start seeing the errors shown in the following image and it doesn't allow me to interact with the console:    I also see these errors for a while everytime I boot. I need to do Ctrl + Alt + F1 to access a non graphical terminal to install some Nvidia drivers. What's going on?  What's causing the problem seems to be: 00:1c.5 PCI bridge [0604]: Intel Corporation Sunrise Point-H PCI Express Root Port #6 [8086:a115] (rev f1)  jpiabrantes@joao:~$ lspci -nn 00:00.0 Host bridge [0600]: Intel Corporation Sky Lake Host Bridge/DRAM Registers [8086:1910] (rev 07) 00:01.0 PCI bridge [0604]: Intel Corporation Sky Lake PCIe Controller (x16) [8086:1901] (rev 07) 00:02.0 VGA compatible controller [0300]: Intel Corporation Skylake Integrated Graphics [8086:191b] (rev 06) 00:04.0 Signal processing controller [1180]: Intel Corporation Skylake Processor Thermal Subsystem [8086:1903] (rev 07) 00:14.0 USB controller [0c03]: Intel Corporation Sunrise Point-H USB 3.0 xHCI Controller [8086:a12f] (rev 31) 00:14.2 Signal processing controller [1180]: Intel Corporation Sunrise Point-H Thermal subsystem [8086:a131] (rev 31) 00:16.0 Communication controller [0780]: Intel Corporation Sunrise Point-H CSME HECI #1 [8086:a13a] (rev 31) 00:17.0 SATA controller [0106]: Intel Corporation Sunrise Point-H SATA Controller [AHCI mode] [8086:a103] (rev 31) 00:1c.0 PCI bridge [0604]: Intel Corporation Sunrise Point-H PCI Express Root Port #5 [8086:a114] (rev f1) 00:1c.5 PCI bridge [0604]: Intel Corporation Sunrise Point-H PCI Express Root Port #6 [8086:a115] (rev f1) 00:1c.6 PCI bridge [0604]: Intel Corporation Sunrise Point-H PCI Express Root Port #7 [8086:a116] (rev f1) 00:1f.0 ISA bridge [0601]: Intel Corporation Sunrise Point-H LPC Controller [8086:a14e] (rev 31) 00:1f.2 Memory controller [0580]: Intel Corporation Sunrise Point-H PMC [8086:a121] (rev 31) 00:1f.3 Audio device [0403]: Intel Corporation Sunrise Point-H HD Audio [8086:a170] (rev 31) 00:1f.4 SMBus [0c05]: Intel Corporation Sunrise Point-H SMBus [8086:a123] (rev 31) 01:00.0 3D controller [0302]: NVIDIA Corporation GM107M [GeForce GTX 950M] [10de:139a] (rev a2) 07:00.0 Unassigned class [ff00]: Realtek Semiconductor Co., Ltd. RTS522A PCI Express Card Reader [10ec:522a] (rev 01) 08:00.0 Network controller [0280]: Realtek Semiconductor Co., Ltd. RTL8723BE PCIe Wireless Network Adapter [10ec:b723] 09:00.0 Ethernet controller [0200]: Realtek Semiconductor Co., Ltd. RTL8101/2/6E PCI Express Fast/Gigabit Ethernet controller [10ec:8136] (rev 0a) What would be an informative introduction to quantum computing software? I am new to Stack Exchange and am working on a quantum learning platform for minority youth groups (LGBTQ, low-income, at risk, etc).  In the question below they are looking for courses on the subject, which I am also interested in, and do plan on checking those links out for ideas.        What I am looking for are simple videos, articles, or even games, that cover basic quantum theory at an introductory level.  There are some games I have looked into and played.  Hello Quantum! was fun and informative, though on my end there was still a lack of comprehension on how the quantum computer (or anything else "quantum") would actually function and play out.   My focus for the educational platform is more directed towards the software side of quantum computing. Is there anything  that gives a good introduction to the functions and uses a quantum computer will have? As well as what language would be best to program one? Also, would there be a way to program a quantum computer through a classical computer? And, is there a simple introduction to any of this already existing? How do I create a PDO parameterized query with a LIKE statement? I am on a multiplayer Minecraft realm in 1.16 and I was wondering if there was any way to set a chest's Items to a player's entire Inventory including the offhand and armor slots. Using data modify &lt;chest coordinates&gt; Items set from entity &lt;player&gt; Inventory, I was only able to copy over the hotbar and the top two rows of a player's inventory. I honestly don't care if it takes 50 command blocks, any response is gladly appreciated. So if I'm correct, when you chose your starter gear and wealth you can either choose to roll some dice depending on your class and buy everything from that or you can choose the starter gear from your class and add the background items (including some gold). Here are my 2 questions:   If I choose the gear from my class, do I have to take all the equipment from only one of the (a) or (b) choices, or can I freely mix-and-match choices from the two sets? Can you spend the extra gold you get from your background on extra equipment (e.g. some soap and some fishing equipment) or do you have to start with that gold in its pouch?   The reason I ask is that I have recently created a level 1 wood elf druid and I mixed-and-matched the gear from the class, so that I chose a shortbow as a "(b) simple weapon" for the first point and a "(a) Scimitar" at the second point. My DM wasn't so happy about this, as he can't understand why I'm mixing (a) and (b).  He also wasn't so happy with me spending my 10gp from my outlander background. I decided that my druid wouldn't lay so much worth in 10gp compared to a bunch of stuff I got for that (e.g. some soap and some fishing equipment and some other useful things) and he seems to think that I'm supposed to just start with the gold instead. Who invented "dooblidoo"? The word  is used by several different youtube channels as a different word for the youtube description bar. I've seen it used by the  and by .  Who was first to use and invent the term? How can I prevent Google mistakenly offering to translate a page? Several of my site's pages are appearing in search results with [Translate this page] next to it. When I click that it takes me to Google Translate and translates my page "from Catalan to English".  The pages are in English but have a couple of foreign words (actually Japanese romanisations, not Catalan) that appear to be tripping Google up.  A few weeks ago I set the html tag to &lt;html lang="en"&gt; which from research appears to be the best method to specify the language of a document. Google has cached the pages with this attribute but it is still offering to translate.  More research led me to a "notranslate" attribute which prevents translation entirely: &lt;html lang="en" class="notranslate"&gt;. The problem now is users cannot translate from English to their desired language!  Are there any other solutions that force Google to parse my site as English only? Does slowing light violate relativity? Easiest way to move my Windows installation to an SSD? I've taken the plunge and bought an SSD and want to move my existing Windows installation over. The current hard disk is 500Gb, but I've trimmed the contents down to about ~40Gb. I'm transferring it across to a 100Gb SSD and looking for the easiest way just to copy everything across and set the SSD up as a boot device.  I've looked at a few tools like Macrium Reflect, but they don't seem able to restore to a smaller drive. Do I need to go for something like  to do this? I'm trying to avoid scary Linux-based boot utilities if possible, does anyone know of an easier way? In reading about the  protocol I noticed how much its proofs-of-work apparently depend on  of the SHA-256 hash function. And so  presumably do many other applications.  How do researchers go about proving uniformity of proposed hash functions? Is it just an empirical matter of running many inputs through the function and observing the distribution of outputs, or can one formally prove the property? Where could one (non-crypto-expert) learn about how such proofs are constructed and look one up e.g. for the case of SHA-256? Print lines between (and excluding) two patterns I'm going to submit form using cURL, where some of the contents is come from other file, selected using sed  If param1 is line matching pattern from other file using sed, below command will works fine:  curl -d param1="$(sed -n '/matchpattern/p' file.txt)" -d param2=value2 http://example.com/submit   Now, go to the problem. I want show only text between 2 matching pattern excluding the matching pattern itself.  Lets say file.txt contains:  Bla bla bla firstmatch It is a long established fact that a reader will be distracted by the readable content of a page when looking at its layout. secondmatch The point of using Lorem Ipsum is that it has a more-or-less normal distribution of letters, as opposed to using 'Content here, content here', making it look like readable English.   Currently, lots of "beetween 2 matching pattern" sed command won't remove firstmatch and secondmatch.  I want the result to become:  It is a long established fact that a reader will be distracted by the readable content of a page when looking at its layout. How to run scripts without typing the full path? Why don't metals bond when touched together? It is my understanding that metals are a crystal lattice of ions, held together by delocalized electrons, which move freely through the lattice (and conduct electricity, heat, etc.).   If two pieces of the same metal are touched together, why don't they bond?   It seems to me the delocalized electrons would move from one metal to the other, and extend the bond, holding the two pieces together. If the electrons don't move freely from one piece to the other, why would this not happen when a current is applied (through the two pieces)? Related:  (not a dupe, that's a feature request and was closed as a dupe of another one)        (old) post does indeed say that Link-only answers can be flagged as NAA, but there are a lot of posts like  which define Not An Answer is a way that it doesn't apply to link-only answers.  Can we get some up-to-date consensus on this? That link only answers are not allowed is . How use spaces instead of underscores in file names? I'm referring to this "feature"   It was not happening before, but it is now, and it is messing up the structure of some folders I previously created. Is there a way to disable it?