I had learned the principle of least action.But I didn't get the motive behind taking the least action. Or why should the particle follow a path where it have a least action?
I'll be generous and say it might be reasonable to assume that nature would tend to minimize, or maybe even maximize, the integral over time of $T-V$.  Okay, fine.  You write down the action functional, require that it be a minimum (or maximum), and arrive at the Euler-Lagrange equations.  Great.  But now you want these Euler-Lagrange equations to not just be derivable from the Principle of Least Action, but you want it to be equivalent to the Principle of Least Action.  After thinking about it for awhile, you realize that this implies that the Principle of Least Action isn't really the Principle of Least Action at all:  it's the "Principle of Stationary Action".  Maybe this is just me, but as generous as I may be, I will not grant you that it is "natural" to assume that nature tends to choose the path that is stationary point of the action functional.  Not to mention, it isn't even obvious that there is such a path, or if there is one, that it is unique.  But the problems don't stop there.  Even if you grant the "Principle of Stationary Action" as fundamentally and universally true, you realize that not all the equations of motions that you would like to have are derivable from this if you restrict yourself to a Lagrangian of the form $T-V$.  As far as I can tell, from here it's a matter of playing around until you get a Lagrangian that produces the equations of motion you want.  From my (perhaps naive point of view), there is nothing at all particularly natural (although I will admit, it is quite useful) about the formulation of classical mechanics this way.  Of course, this wouldn't be such a big deal if these classical ideas stayed with the classical physics, but these ideas are absolutely fundamental to how we think about things as modern as quantum field theory.  Could someone please convince me that there is something natural about the choice of the Lagrangian formulation of classical mechanics (I don't mean in comparison with the Hamiltonian formulation; I mean period), and in fact, that it is so natural that we would not even dare abandon these ideas?
Most efficient way to check for DBNull and then assign to a variable? Proving a direct product of groups is a group I am trying to prove the following and am looking toward the math.stackexchange community to comment on whether I am on the right track or not. Thank you in advance.    Let $n \geq 1$ be an integer. Suppose $\{G_1,...,G_n\}$ is a finite set of groups. Define the set $G = G_1 \times G_2 \times ... \times G_n$ of ordered n-tuples $(g_1,g_2,...,g_n)$ where $g_i \in G_i$. Show that the operation $\bullet$: $G \times G \rightarrow G$ given by $(g_1,g_2,...,g_n) \bullet (g'_1,g'_2,...,g'_n) = (g_1g'_1,g_2g'_2,...,g_ng'_n)$ makes $G$ into a group.    Proof of associativity: $((g_1,g_2,...,g_n) \bullet (g'_1,g'_2,...g'_n)) \bullet (g''_1,g''_2,...,g''_n)$ = $(g_1g'_1,g_2g'_2,...,g_ng'_n) \bullet (g''_1,g''_2,...,g''_n) $ = $(g_1g'_1g''_1,g_2g'_2g''_2,...,g_ng'_ng''_n)$ = $(g_1,g_2,...,g_n) \bullet (g'_1g''_1,g'_2g''_2,...,g'_ng''_n) = (g_1,g_2,...,g_n) \bullet ((g'_1,g'_2,...,g'_n) \bullet (g''_1,g''_2,...,g''_n))$  \  Proof of the existence of an identity For each $G_i \in G$, take its identity element $e_i$ and multiply it as follows: $(g_1,g_2,...,g_n) \bullet (e_1,e_2,...,e_3)$ = $(g_1,g_2,...,g_n)$.  \  Proof of the existence of an inverse: Since each $G_i \in G$ is a group, simply takes its multiplicative inverse $g_i^{-1}$ and multiply it: $(g_1,g_2,...,g_n) \bullet (g_1^{-1},g_2^{-1},...,g_n^{-1}) = (e_1,e_2,...,e_n)$. Is there a way to "hide" a normal windowed application in Windows?  I'm looking for something that removes it from the taskbar and possibly puts an icon in the system tray instead. Managing Metered Connections on OSX WiFi not working Kubuntu 20.04 - Intel 8265 Network Interface Real Analysis, Folland Problem 1.3.15 Measures Given a measure $\mu$ on $(X,M)$, define $\mu_0$ on $M$ by $$\mu_0(E) = \sup\{\mu(F): F\subset E \ \text{and} \ \mu(F) &lt; \infty\}$$ a.) $\mu_0$ is a semifinite measure. It is called the semifinite part of $\mu$. b.) If $\mu$ is semifinite, then $\mu = \mu_0$ (Use Exercise 14) c.) There is a measure $\nu$ on $M$ (in general, not unique) which assumes only the values of $0$ and $\infty$ such that $\mu = \mu_0 + \nu$.  The proof of Exercise 14 can be found  For a.) I believe we need to show that $\mu_0(E) = \infty$. Perhaps we can do this by proof of contradiction, similarly to what the proof in the link above does (actually it seems identical). For b.) I am not sure how we can show $\mu = \mu_0$ I think maybe we have to show by construction that $\mu \leq \mu_0$ and $\mu \geq \mu_0$ For c.) I haven't any idea for. Any suggestions is greatly appreciated. Until I went to army, I (had been / used to be / was ) a introverted person. - which one is right? Until I went into the army, I (had been / used to be / was ) an introverted person. And after I I finished army service, I became more confident.   I think all of these are possible.  As for had been, I'm referring to the state of me being introverted before the accident that happened in the past. So, I think I can use had been.  And at the same time I feel I can use used to be and was.. If I don't want to emphasize the time gap between two accident, I feel I don't need to use past perfect. So I think this problem depends on how I'm going to say something.   which one is preferable? and what is the difference? 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 have just bought a Nikon 50mm f/1.8D prime lens.  I took this picture (leg of my glasses frame on a mouse, and a LED desk lamp is on top of it) : f1.8, 1/80s, ISO 200   The purple fringing is right in the middle of the picture. Does it happen very often to this lens? How do I show that $x^4-10x^2+1$ is irreducible over $\mathbb{Q}$? Someone says I should use the rational root test, but I don't exactly know how that applies. Thanks for any input. Tips to extend battery life for laptops and notebooks This question is present as a matter of historical interest.  While you are encouraged to help maintain its answers, please understand that "big list" questions are not generally allowed on Ask Ubuntu and will be closed per the .   It is possible to greatly extend the usage time of a laptop or netbook running on battery by disabling various services and installing various packages.   What tricks or tips do people have for getting an extra hour or two out of their batteries? Einstein's first postulate implies the second? for working with MS word files in python, there is python win32 extensions, which can be used in windows. How do I do the same in linux? Is there any library? Could anyone provide an example of a nonempty subset $U$ of $R^2$ such that $U$ is closed under scalar multiplication, but $U$ is not a subspace of $R^2$ math fallacy problem: $-1= (-1)^3 = (-1)^{6/2} = \sqrt{(-1)^6}= 1$? I know there is something wrong with this but I don't know where. It's some kind of a math fallacy and it is driving me crazy. Here it is: $$-1= (-1)^3 = (-1)^{6/2} = \sqrt{(-1)^6}= 1?$$ Understanding backtick (`) Index entries duplicated for captions In document class book, and package caption, creating index with makeidx. Index entries for the main text and for footnotes collate properly; but those for captions create their own entries.  Example: for the tag \index{New York!Custom House!\textit{Cartouche}}, text and footnotes collate properly; but used in a caption, a new entry is created at the third level, and is also out-of-alphabetic-sequence. Note: the top level, and first sub are correct; only the second sub is duplicated.  How to get the caption index entries under the same entry? From what I understand, the frequency of light coming from the source moving towards an observer increases. From $ E=hv $ , this implies increase in energy of photon.  What really is confusing, is where does that extra energy come from? And also where is the energy is lost during opposite Doppler effect (red shift)? Why doesn't this violate the conservation of energy? Why do we use the base form of verbs instead of the past form in past tense questions? In regard to this , my question is similar but that answer is not clear. I want to know why we use base form of verb, e.g. 'go' to form the past tense instead of past form such as 'went'?     Question: Did you go to a bank?  (Why we use 'go' here, rather than 'went'?)   Answer: I went to a bank.   Am I confusing verb forms with tenses? Please note that I'm not a native speaker so some clarification or reference on this would be helpful. WUBI installed Ubuntu not showing up at boot menu after Windows reinstallation