Hamilton's Principle is that the first order variations of $\displaystyle\int^{t_2}_{t_1} L$ $ dt$ for an on-shell trajectory in the configuration space should vanish provided the varied off-shell trajectories meet the on-shell trajectory at $t_1$ and $t_2$.   The only reason I can think of as to why this should not be called the Principle of Least Action is that it is not - it is the Principle of Stationary Action. But leave that distinction aside for a moment because it is sort of irrelevant for the doubt that I have, as will become clear in the rest of my post.   The issue that I have is that my Professor is insisting that the condition obtained on the ''$\Delta$ - variation" of $\displaystyle\int^{t_2}_{t_1} L$ $ dt$ about an on-shell trajectory in the configuration space for a system with conserved Hamiltonian (both on-shell and off-shell) is exclusively what should be called the Principle of Least Action. This condition is that $$\Delta \displaystyle\int^{t_1}_{t_2}p\dot q dt=0$$ where the $\Delta$-variation is defined as the following:  $$\Delta \displaystyle\int^{t_1}_{t_2}p\dot q dt=\displaystyle\int^{t_1+\Delta t_1}_{t_2 + \Delta t_2}p\dot q dt + \delta \displaystyle\int^{t_1}_{t_2}p\dot q dt$$ with a key difference from the variations considered in the Hamilton's principle that apart from varying the limits of integration as well, we do not require the varied off-shell trajectories to coincide with the on-shell trajectory at the end-points of the on-shell trajectory.   Now, I have always read that $\displaystyle\int L dt$ is what is called action and thus, I think that a variational principle associated with the same should be called the Principle of Least (Stationary) Action. Whereas $\displaystyle\int p\dot q dt$ defer from $\displaystyle\int L dt$ by $\displaystyle\int H dt$ - jeopardizing the possibility of its variational principle's to be called "Principle of Least (Stationary) Action" (leave alone an exclusive right to be called so).
What are the difference between:    the action $\int_{t_{1}}^{t_{2}}(L+H) dt$ that we use in the principle of least action, and  the action integral $\int_{t_{1}}^{t_{2}}L dt$ that we use in Hamilton's variational principle?   References:   H. Goldstein, Classical Mechanics, Section 8.6. H. Goldstein, Classical Mechanics, Chapter 2.
After installing Telegram Desktop on Ubuntu 16.04 by these commands:  sudo add-apt-repository ppa:atareao/telegram sudo apt-get update sudo apt-get install telegram   I get this error:  $ telegram telegram: Panic - Unable to find your tty (pts/2) in /var/run/utmp What is the meaning of the third derivative of a function at a point (Originally  by AJAY.)  What is the geometric, physical, or other meaning of the third derivative of a function at a point?  If you have interesting things to say about the meaning of the first and second derivatives, please do so. LuaLaTeX protrusion problem with Negative value Prove that for every $a\in A$ $\exists$ $q\in M,s\in A$ s.t. $a=sba+qa$ where $M$ is maximal ideal in $A$. Let A be a commutative ring with unity ring, M be a maximal ideal and $b\notin M$ . Show that there exists $q \in M, s\in A$ such that for every $a \in A$ we have $a=sba+qa$. Show that this is true only when $M$ is a maximal ideal.   Suppose if we take $A=2\mathbb Z$ and $M=4\mathbb Z$, $a=6.$ Then by the given statement $\exists$ $s\in A$, $q\in M$ s.t. $6=6(2s+q)$ $\implies$$s=\frac{1-q}{2}\implies q \text { is odd}$. But $q \in 4\mathbb Z.$  Please someone give some hints. how can solve this.  Thank you.. Is there a location where I can drop in a custom JavaScript file in Drupal 7.x?  It just includes simple event actions for menus, and I don't want to edit the theme/module files.  Or do I have to just add the "include" hook to the theme/module? Difference between "does have" and "has" Possible Duplicate:                 What is the difference between does have and has? For example, compare she does have a car and she has a car. How do I list all of the files in a commit? I am looking for a simple Git command that provides a nicely formatted list of all files that were part of the commit given by a hash (SHA-1), with no extraneous information. I have tried: git show a303aa90779efdd2f6b9d90693e2cbbbe4613c1d  Although it lists the files, it also includes unwanted diff information for each. Is there another git command that will provide just the list I want, so that I can avoid parsing it from the git show output? I need to insert 800000 records into an MS Access table. I am using Delphi 2007 and the TAdoXxxx components. The table contains some integer fields, one float field and one text field with only one character. There is a primary key on one of the integer fields (which is not autoinc) and two indexes on another integer and the float field.  Inserting the data using AdoTable.AppendRecord(...) takes > 10 Minutes which is not acceptable since this is done every time the user starts using a new database with the program. I cannot prefill the table because the data comes from another database (which is not accessible through ADO).  I managed to get down to around 1 minute by writing the records to a tab separated text file and using a tAdoCommand object to execute  insert into table (...) select * from [filename.txt] in "c:\somedir" "Text;HDR=Yes"   But I don't like the overhead of this.  There must be a better way, I think.  EDIT:  Some additional information:   MS Access was chosen because it does not need any additional installation on the target machine(s) and the whole database is contained in one file which can be easily copied. This is a single user application. The data will be inserted only once and will not change for the lifetime of the database. Though, the table contains one additional field that is used as a flag to indicate that the corresponding record in another database has been processed by the user. One minute is acceptable (up to 3 minutes would be too) and my solution works, but it seems too complicated to me, so I thought there should be an easier way to do this. Once the data has been inserted, the performance of the table is quite good. When I started planning/implementing the feature of the program working with the Access database the table was not required. It only became necessary later on, when another feature was requested by the customer. (Isn't that always the case?)   EDIT:  From all the answers I got so far, it seems that I already got the fastest method for inserting that much data into an Access table. Thanks to everybody, I appreciate your help. What do you call a person who started something that is later followed by everyone? Create a bootable USB on a Mac to install Ubuntu on a PC Distribution of Dark Matter around galaxies It is well-known that measurements of the velocity profile in galaxies are not compatible with Newtonian laws. A way to circumvent the problem is to assume that galaxies are surrounded by a spherical halo of Dark Matter. The density of mass of this Dark Matter can then be computed from the velocity profile with Newton's laws.   My question is the following: the formation of a galaxy involves essentially only the gravitational interaction. According to Newton (and General Relativity too), the trajectory of a particle does not depend on its mass. As a consequence, I do not understand how usual matter and Dark Matter could have followed different trajectories leading to such different mass distributions nowadays. Can somebody explain this (apparent) paradox ? Can I still use logistic regression if some variables are not applied to all the observational units? Some of the variables are questions that are not being asked to every single person in the questionaire depending on how they answered previous question(s). So would I still be able to use a logistic regression if some of the questions (the independent variables) are only measuring the responses of SOME of the respondents? Use /testfor to make a custom command Basically, I'm trying to recreate the Faction plugins /f map in my single-player without mods as my PC doesn't run them very well. (I know it sounds strange.) I am trying to use the /testfor to test if I have written '/f map' in chat, and if I have, it would show me a custom generated map. I don't know how to word this better xD Essentially what I want is for it to test if I have written a command in chat, and if I have, display something. Why do return flight costs vary depending on the country of origin? Creating Inky-looking characters in Blender I'm involved in a project to do with ink animation and experimentation. I found this incredible piece of animation:    I originally thought this was some sort of After Effects compositing (is it even possible in AE?!), but found out that this particular animation was created in 3DS Max with Krakatoa and FumeFX. I'm just wonder if a similar effect is doable with Blender's smoke simulator.  There are various YouTube videos that have pretty decent results of ink renders from Blender, but I can't seem to find anything of a similar sort of movement animation. Just blobs floating around! The closest thing I found was this animation of a         but again I can't seem to replicate this. The smoke just doesn't seem to follow my character's animation. Any help with this would be greatly appreciated. Thanks. What is Chromatic Aberration? Is it a physical part of the lens, or just an optical illusion? Elementary formula for permutations? What should I be teaching my two year old? My two year old daughter has never been to a daycare, so everything she knows is something that me, my husband, or her grandparents have taught her. She sees very little interaction with other kids her age, well, except for her 8 month old sister.   I am wondering what people, or even day care centers, teach two year olds. What do they do all day long? Presently, our daughter draws a lot - she has bathtub crayons, wax crayons for paper, and chalk for the driveway/sidewalk. She attempts to draw circles and lines. We try to teach her shapes and colors, and she has a rough idea of what they are. She plays with sand in the sandbox a lot, dumping sand all over her and her sister's head. She goes to her playhouse and uses the slide and the swing.   We sing songs together - the alphabet song or the numbers song. She is able to say the correct word when I say the phonic (i.e. if I say B, she'll say ball; if I say H, she'll say hat). She tries counting, but is still learning (she goes 1, 2, 6, 7, 8...). She enjoys building and destroying towers. We read books and I sing some nursery rhymes. She plays a toy piano and dances around to it. She plays with playdough, but most of the time it just ends up in crumbs everywhere.  She also watches TV - something I know most parents object to. She enjoys watching Dora, Finding Nemo, and Cinderella. She will sometimes watch LeapFrog phonics videos for alphabet and numbers.   I am not trying to give a resume for my daughter, but merely wondering is there something more that kids her age do? Is there something else that I should be teaching her? What do kids her age do at a daycare?   Thanks A random variable that induces a $\sigma$-algebra the same as the one in the sample space Do all accepted students in top-tier engineering programs get invited to interview before getting admitted?