If we start from  $ m\ddot{x} = -\frac{dV}{dx} $  How could one derive/construct principle of least action? i.e. find out that this quantity  $ S = \int_{t_0}^{t_1}dt\left(\frac{m\dot{x}^2}{2} - V(x) \right)$  needs to be minimized. Assuming it's true and then proving it is is very easy and it's written in every textbook but that's not what I'm asking for. I want to go in other direction, not assuming I know the answer. All I know are Newton's laws and that $ F = -dV/dx$.  I thought of simply inverting the steps of derivation of Euler-Lagrange equation but they seem unintuitive and the reasoning behind them is unclear except that I know the answer in advance (which I want to assume that I don't).  I guess this is inverse of optimization problem? Given a solution finding out what does it optimize?
Is there a proof from the first principle that for the Lagrangian $L$,    $$L = T\text{(kinetic energy)} - V\text{(potential energy)}$$  in classical mechanics? Assume that Cartesian coordinates are used. Among the combinations, $L = T - nV$, only $n=1$ works. Is there a fundamental reason for it?  On the other hand, the variational principle used in deriving the equations of motion, Euler-Lagrange equation, is general enough (can be used to to find the optimum of any parametrized integral) and does not specify the form of Lagrangian. I appreciate for anyone who gives the  answer, and if possible, the primary source (who published the answer first in the literature).      Notes added on Sept 22: - Both answers are correct as far as I can find. Both answerers were not sure about what I meant by the term I used: 'first principle'. I like to elaborate what I was thinking, not meant to be condescending or anything near to that. Please have a little understanding if the words I use are not well-thought of. - We do science by collecting facts, forming empirical laws, building a theory which generalizes the laws, then we go back to the lab and find if the generalization part can stand up to the verification. Newton's laws are close to the end of empirical laws, meaning that they are easily verified in the lab. These laws are not limited to gravity, but are used mostly under the condition of gravity. When we generalize and express them in Lagrangian or Hamiltonian, they can be used where Newton's laws cannot, for example, on electromagnetism, or any other forces unknown to us. Lagrangian or Hamiltonian and the derived equations of motion are generalizations and more on the theory side, relatively speaking; at least those are a little more theoretical than Newton's laws. We still go to lab to verify these generalizations, but it's somewhat harder to do so, like we have to use Large Hadron Collider. - But here is a new problem, as @Jerry Schirmer pointed out in his comment and I agreed. Lagrangian is great tool if we know its expression. If we don't, then we are at lost. Lagrangian is almost as useless as Newton's laws for a new mysterious force. It's almost as useless but not quite, because we can try and error. We have much better luck to try and error on Lagrangian than on equations of motion. - Oh, variational principle is a 'first principle' in my mind and is used to derive Euler-Lagrange equation. But variational principle does not give a clue about the explicit expression of Lagrangian. This is the point I'm driving at. This is why I'm looking for help, say, in Physics SE. If someone knew the reason why n=1 in L=T-nV, then we could use this reasoning to find out about a mysterious force. It looks like that someone is in the future.
How to set BibTeX bibliography title? Currently my bibliography says "Literatur" (german). But I want something completely different there. How can I achieve that? Solder does not stick to tip of soldering iron? Basic question here, but it's driving me insane!  The solder does not seem to stick to the tip of my soldering iron. After the iron heats up, the tip appears black (sometimes on all sides and sometimes on one side), then either it doesn't melt the solder at all, or it I fiddle around with it for a while; it melts the solder, but the solder rolls into itself into a ball and doesn't stick to the tip.  I've seen this many times while soldering before and usually it's solved by cleaning the tip on a soldering sponge or cooling it down and scraping the tip off with a blade, but I have scraped this particular one many times and still it's the same problem: it heats up and gets black, then the solder doesn't stick to it.  It doesn't help that this soldering iron has a very fine tip and I have to use it to solder very tiny chips and wires under a microscope (I'm just starting to learn soldering under microscopes).  Any ideas on what I can do?    PS: Is scraping the tip a bad idea? I have been doing it with my other bigger soldering irons and it seemed to work (at least in most cases) but I am not sure whether it is a good solution or not. Can inductor voltage and capacitor current change abruptly? I understand that inductor current and capacitor voltage cannot change abruptly, but can inductor voltage and capacitor current change abruptly?  I have a feeling the answer is no but I cannot explain why. How can I bring STAG back if I have 100% completion? There's a couple of the Saintsbook challenges that require you to have STAG forces available to kill.  After you complete the game, however, the STAG forces are nowhere to be found.  I believe that you can fight a few more STAG forces if you replay the final mission, but I think my save is glitched, so that's not an option.    Is there any way post-game completion to bring STAG back?  I miss their much nicer tanks and VTOLs for hijacking :( Wielding two weapons without "Two-Weapon Fighting" or "Dual Wielder"? We have a Barbarian on our team that very much likes to switch up his weapons; he goes from a Dwarven Hammer, to a pair of Hand-Axes, to occasionally just going bare-fisted. No issues there, he's fairly adept with all 3.  However, I have only just recently picked up on the way that he rolls his attacks for the two Hand-Axes: He rolls the first attack, adds his modifiers etc. Then he rolls the second attack, this time without modifiers, which is all well and good. But I'm not sure if this action uses his bonus action or not, or is considered a separate action (though, being a level 4 Barbarian, he only has one action per turn), or is merely part of the same action.  If we have a look at the Two-Weapon Fighting feature from the Fighter's fighting styles:     Two-Weapon Fighting   When you engage in two-weapon fighting, you can add your ability modifier to the damage of the second attack. (PHB5e p.72)   Similarly, the Dual Wielder feat also uses the Two-Weapon Fighting feat:     Dual Wielder   You master fighting with two weapons, gaining the following benefits:      ...         You can use two-weapon fighting even when the one-handed melee weapons you are wielding aren’t light. (PHB5e p.164, emphasis mine)      So; yes, this gives a bonus to the second weapon attack, but having asked a  to clarify which action this secondary weapon attack belongs to, it was determined that this uses your bonus action;     [...] the second attack requires a bonus action   So after that long winded explanation, here is the question: Without any additional feats, can you wield and use two weapons at once, and if so, how many actions does it require to make a full attack with both weapons in a turn? Showing margins/frames in beamer slides Word for the situation of being unable to pass opposing pedestrian, as you both start to step same direction Is there a word or expression in English, which describes the situation, when you can't pass a  stranger, who is walking towards you on the street, because you both start to step the same direction?  I'm pretty sure I've heard one once. As I remember, it was including "dancing" reference, but in more specific way. For a given inner product space $V$, I would like to prove the inequality $$\|\mathbf{x}+\mathbf{y}\|^2 + \|\mathbf{x}-\mathbf{y}\|^2 = 2(\|\mathbf{x}\|^2 + \|\mathbf{y}\|^2).$$  We see that $$\|\mathbf{x}+\mathbf{y}\|^2 + \|\mathbf{x}-\mathbf{y}\|^2 = \langle\mathbf{x}+\mathbf{y},\mathbf{x}+\mathbf{y}\rangle + \langle\mathbf{x}-\mathbf{y},\mathbf{x}-\mathbf{y}\rangle,$$  which by linearity $$=\langle\mathbf{x},\mathbf{x}+\mathbf{y}\rangle + \langle\mathbf{y},\mathbf{x}+\mathbf{y}\rangle + \langle\mathbf{x},\mathbf{x}-\mathbf{y}\rangle - \langle\mathbf{y},\mathbf{x}-\mathbf{y}\rangle.$$  I've been trying to figure out how to go about this proof using linearity in the second argument of an inner product, but my textbook does not say that linearity necessarily holds in the second component. If it does, how would I go about the rest of this proof? If not, should I be potentially using some aspect of conjugate symmetry to prove this statement? Which year does Spider-Man: Homecoming take place in? The first scene of the movie is presumably set in the immediate aftermath of The Avengers. The movie then fast-forwards to '8 years later'.  But Spider-Man already appeared in Civil War as a high school student, which he still is in this movie. So what's the timeline here? If Avengers was set in 2012 and Civil War in 2016, how can Homecoming be in 2020?  Or am I wrong in assuming the canonical timelines of the other movies? Clarification of the "25D" option in ogr2ogr Data extension values in the "Update Contact" activity in Journey Builder I'm looking for a way to use fields from a data extension (that is the entry source for a journey) in the "Update Contact" activity.  I want to create a sort of log in a different data extension. Let's say I have a sourceDEthat is used by the journey and a journeylogDE where I would like to write the email of contact that's currently in the journey.  What should I write in the value field to get a value from the currently used record in the sourceDE?  I tried {{Contact.Attribute.sourceDE.email}}, but it did not return any values.   I would appreciate it if you could give me any tips how to fix this issue. Dealing with questions of nefarious intent How should we treat questions that are, or have seeds of, evil or malicious purposes and actions, such as writing spambots and malware?    Should they be answered like regular "everybody keeps their pants on" questions? Or should there be some other type of response to this bad behaviour lot? We don't get much snow in the UK, so I don't have much experience photographing it, and when I tried photographing falling snow last year I found it very hard to capture the snowflakes in the air — they either didn't show up or were just streaks. Is there a shutter speed that is best for this? I don't want to use flash to do so.  Any other snow photography tips welcome from those in snowier places! Error with square bracket in table \begin{table}[ht] \makebox[\textwidth][c]{     \begin{tabular}{ccccc}         \hline         \hline         [] &amp; [] &amp; [] &amp; [] &amp; []\\         [] &amp; [] &amp; [] &amp; [] &amp; []\\         \hline         \hline     \end{tabular} } \caption{test} \end{table}   I cannot run the above code, latex returned an error about missing number, but I can't see anything I am missing.  And the weirdest thing is that  \begin{table}[ht] \makebox[\textwidth][c]{     \begin{tabular}{ccccc}         \hline         \hline         [] &amp; [] &amp; [] &amp; [] &amp; []\\         \hline         \hline     \end{tabular} } \caption{test} \end{table}   This code works.  Anyone have any idea? If I type something like this:  {\itshape\LARGE   Dunt venim dolorerosto do odit   ametum vulla conum dolore   conulput in vero od el   Dunt venim dolorerosto do odit   ametum vulla conum dolore   conulput in vero od el }   it results in a block of 4 lines with narrow spacing between the 1st and 2nd line and between the 3rd and 4th line. Line spacing between the 2nd and 3rd line is normal.  I can't fully explaing it to myself so I am looking for a short description for this behaviour. I know for example that with a \par at the end of the text (inside the group), the calculation would correct. But I have no idea why... Quantum entanglement faster than speed of light? recently i was watching a  on quantum computing where the narrators describes that quantum entanglement information travels faster than light!  Is it really possible for anything to move faster than light? Or are the narrators just wrong?  Regards, Is there a word for a question asked despite knowing the answer, but meant to elicit a response for the sake of the audience? In journalistic writing I often see writers, correspondents, and interviewers use questions in non-literal ways.  Of course there are rhetorical questions designed to make a point and not meant to literally elicit an answer.   Instead of that, I have in mind when an interviewer asks a correspondent, "why did she not go to the police?" when the interviewer already knows the story and knows the answer to the question, but wants the correspondent to articulate the answer for the audience's benefit.    Is there a commonly used term for this sort of a question?  A "leading" question generally implies an attempt to manipulate the respondent into giving certain responses, which is not exactly the same as what I'm describing. Convert a JSON string to object in Java ME? What are "schlieren" in English? The German word for these patterns is "Schliere, pl. Schlieren", and apparently this word has entered the English language as a loanword (cf. article  in Wikipedia or entry  in OED). But the German loanword is probably not something most native speakers of English understand without the help of a dictionary, so what would you call, or how would you describe, the following patterns?  Concrete:    Air:    Oil on water:    Coffee and milk:    Paint:    Smoke:    Rock:    Marbled paper:    And so on.    "Waves" is not applicable, because waves are vertical undulations of the surface, while these are a pattern on, or inside, the surface level, with no amplitude.  "Striæ" does not really fit either, because those refer more to the parallel layers, than to the curly forms. How do you copy a record in a SQL table but swap out the unique id of the new row?