I'm aware that you can always approximate a potential by a quadratic term. But is this the most 'fundamental' reason for the pervasiveness for oscillations?
I've been wondering what makes the  such an important model. What I came up with:   It is a (relatively) simple system, making it a perfect example for physics students to learn principles of classical and quantum mechanics. The harmonic oscillator potential can be used as a model to approximate many physical phenomena quite well.   The first point is sort of meaningless though, I think the real reason is my second point. I'm looking for some materials to read about the different applications of the HO in different areas of physics.
Install Java JDK and JRE on Debian 8 Why does my iPhone power off when the battery is down to 30%? Today I was walking using my iPhone 6S running iOS 9.3 as a pedometer and it shutdown.   When I got home and plugged it in, it said 30% capacity.   That seems strange. I would expect it for 2% or 5% to preserve RAM, but 30%? Maybe there is a setting I don't know about.   My question is: Why does my iPhone power off when the battery is down to 30%?   Edit: this is not a duplicate of the low battery crash as I witnessed the low power shutdown and it was not a crash. I've asked several questions about 4th Edition D&amp;D because I may be able to start a game. However, I'd need to purchase (some) materials. I am aware of the 4 book set (core) and the expected Essentials Red Box.  So now I need to know: presumably buying either the 4 book set or essentials will be enough to get my game on. Would there be something 'lacking' though? Would I be missing character types, for instance, as I've seen that the splatbook strategy is still in effect?  I'm an experienced player and GM and plenty creative, so many holes I'd be able to fill in myself. I just don't want to get sucked into too much financial impact. Passing a dictionary to a function as keyword parameters I'd like to call a function in python using a dictionary.  Here is some code:  d = dict(param='test')  def f(param):     print(param)  f(d)   This prints {'param': 'test'} but I'd like it to just print test.  I'd like it to work similarly for more parameters:  d = dict(p1=1, p2=2) def f2(p1, p2):     print(p1, p2) f2(d)   Is this possible? How does ZFC describe addition? Surprisingly, the Wikipedia article on addition doesn't contain the answer. I looked  elsewhere online for it, but didn't find it.   Intuitively, the cardinal of the union of two sets seemed appealing. But that approach doesn't work, given that the union of 2 and 2 is 2. I wondered whether ordered sets might be essential to the answer.   So, how does ZFC describe addition? Login failed for user - Error 18456 - Severity 14, State 38 Prove that $\mathrm{Res}[f/g,z_0]=\frac{f(z_0)}{g'(z_0)}$ I need to prove that if $f$ and $g$ are analytic in $D_r(z_0)$ and $g$ has a simple zero at $z_0$, then $$\mathrm{Res}[f/g,z_0]=\frac{f(z_0)}{g'(z_0)}$$   When $f(z_0)\neq 0$ and since $1/g$ has a simple pole at $z_0$ $$\mathrm{Res}[f/g,z_0]=\lim_{z\to z_0}(z-z_0)\frac{f(z)}{g(z)}=\frac{f(z_0)}{g'(z_0)}$$ But what do I need to do when $f(z_0)=0$? Any help will be appreciated thanks Syntactic characterisation of the intersection of CTL and LTL As a mod when should I vote to close content that is not flagged? I know mods are supposed to step in to quickly close seriously problematic posts and otherwise allow the community to vote to close a post.  But what about when there are four close votes on a question, and I personally feel the question should be closed but isn't worthy of mod super-mega closing? Should I cast the final vote like a normal user (even though it's technically binding) or should I sit back and wait  for the community?  For me the problem is complicated because on UX.SE we hover close to the point where there's not enough high-rep activity to quickly close posts and I was one of those high rep users that used to VtC.  Should I always treat my mod close vote as binding and reserve it for extreme situations, or should I treat my close vote as "normal" when it's the fifth and final vote anyway? Automatically turn on VPN on computer unlock Ubuntu 18.04 When I wake the laptop from the sleep I need to get the VPN state as I left it (turned on). Can't find the setting of autoconnect in VPN settings... Need some easy way to ask the system to autoconnect the VPN when the internet is available.  Edit based on comments:  Ubuntu 18.04 has no option to Always connect to VPN when using using this connection. Feed randomly gone on SE Android app SE Android App Version: 1.0.84  There is a display issue in the Feed section. The Earlier today, the below sections and the post title are not visible.  There is no exact steps to reproduce. It doesn't happen when the app is initializing. But if you open the Stack Exchange app, navigate to some other apps, then come back after few minutes, I can see this issue (but not all the time).  When the issue is there, if I manually click the Feed option in the navigation sidebar, the display issue is fixed.  I have recorded the issue: I tried to use GNOME Nightlight in Ubuntu 17.04. The settings only allow me to turn it on and adjust the time. Is it possible to adjust the intensity or temperature of the hue? Currently, it's a bit too much. The 'ok' on the 'Your Answer' dialog on unregistered sites is a bit too high The 'ok' on the 'Your Answer' dialog on unregistered sites is a bit too high:     You can see the top of the k is outside the actual box, increasing the margin-top by 10px in the containing div fixes it. Why can't a CNAME record be used at the apex (aka root) of a domain? This is a  about CNAMEs at the apices (or roots) of zones   It's relatively common knowledge that CNAME records at the apex of a domain are a taboo practice.  Example: example.com. IN CNAME ithurts.example.net.  In a best case scenario nameserver software might refuse to load the configuration, and in the worst case it might accept this configuration and invalidate the configuration for example.com.  Recently I had a webhosting company pass instructions to a business unit that we needed to CNAME the apex of our domain to a new record. Knowing that this would be a suicide config when fed to BIND, I advised them that we would not be able to comply and that this was bunk advice in general. The webhosting company took the stance that it is not outright forbidden by standard defining RFCs and that their software supports it. If we could not CNAME the apex, their advice was to have no apex record at all and they would not provide a redirecting webserver. ...What?  Most of us know that  insists that A CNAME record is not allowed to coexist with any other data., but let's be honest with ourselves here, that RFC is only Informational. The closest I know to verbiage that forbids the practice is from :     If a CNAME RR is present at a node, no other data should be   present; this ensures that the data for a canonical name and its aliases   cannot be different.   Unfortunately I've been in the industry long enough to know that "should not" is not the same as "must not", and that's enough rope for most software designers to hang themselves with. Knowing that anything short of a concise link to a slam dunk would be a waste of my time, I ended up letting the company get away with a scolding for recommending configurations that could break commonly used software without proper disclosure.  This brings us to the Q&amp;A. For once I'd like us to get really technical about the insanity of apex CNAMEs, and not skirt around the issue like we usually do when someone posts on the subject.  is off limits, as are any other Informational RFC applicable here that I didn't think of. Let's shut this baby down. How do I use sudo to redirect output to a location I don't have permission to write to? I know that there is no vector space having precisely $6$ elements. Does every set have a group structure? C++ virtual function from constructor Give an example of three positive integers m, n, and r Ball motion with air resistance coupled differential equation for fourth-order Runge-Kutta I've created a MATLAB function for solving coupled differential equation with the fourth-order Runge-Kutta method based on what is provided  (Simultaneous Equations of First Order). Here the function:  function [y] = rk4_c(f, g, h, x, y, z, n) % Runge-Kutta % Implementation of the fourth-order method for coupled equations % h = dt % x is the time here  for ii=1:(n-1)     k1 = h * f(x(ii), y(ii), z(ii));     l1 = h * g(x(ii), y(ii), z(ii));     k2 = h * f(x(ii) + 0.5*h, y(ii) + 0.5*k1, z(ii) + 0.5*l1);     l2 = h * g(x(ii) + 0.5*h, y(ii) + 0.5*k1, z(ii) + 0.5*l1);     k3 = h * f(x(ii) + 0.5*h, y(ii) + 0.5*k2, z(ii) + 0.5*l2);     l3 = h * g(x(ii) + 0.5*h, y(ii) + 0.5*k2, z(ii) + 0.5*l2);     k4 = h * f(x(ii) + h, y(ii) + k3, z(ii) + l3);     l4 = h * g(x(ii) + h, y(ii) + k3, z(ii) + l3);     y(ii+1) = y(ii) + (1/6)*(k1 + 2*k2 + 2*k3 + k4);     z(ii+1) = z(ii) + (1/6)*(l1 + 2*l2 + 2*l3 + l4); end   I would like to use this function for solving motion equations for a ball with air resistance. So starting with the basic equations (I hope they are correct): $$md^2x/dt^2 = F_{D_x}$$ and $$md^2y/dt^2 = F_{D_y}-mg$$ with $$F_{D_x} = -Dvvcos(\theta)$$ and $$F_{D_y} = -Dvvsin(\theta)$$ where $D$ is a drag coefficient. From my understanding, and from what I read, both second-order equation can be converted to two coupled equations that could be used with my coupled Runge-Kutta method function. Can someone help me understand how to convert my equation to a set of coupled equations that could be used with my MATLAB function?  EDIT  I was able to use my rk4_c function, however it only works for solving the velocities. I'm now trying to see how I could solve for the positions x and y. I'm kind of learning about Runge-Kutta methods at the same time, so sorry for the questions that may sound trivials.  % Runge Kutta code to solve projectile motion with quadratic drag % dVx/dt = -(D/m)*vx*sqrt(vx^2+vy^2) % dVy/dt = -(D/m)*vy*sqrt(vx^2+vy^2) - g  clc clear all  % Constant D = 0.24; % m = 2; % kg g = 9.80665; % m/s^2  % Define function handles fVx = @(t,vx,vy) -(D/m)*vx*sqrt(vx^2+vy^2); fVy = @(t,vx,vy) -(D/m)*vy*sqrt(vx^2+vy^2) - g;  % Initial conditions v0 = 200; % m/s theta = 30*pi/180; % rad t(1) = 0; vx(1) = v0*cos(theta); vy(1) = v0*sin(theta);  % Step size h = 0.01; % s tFinal = 2; N = ceil(tFinal/h);  % RK4 simultaneous coupled loop for ii = 1:N     % Update time     t(ii+1) = t(ii) + h;      % Update vx and vy     k1 = h * fVx(t(ii), vx(ii), vy(ii));     l1 = h * fVy(t(ii), vx(ii), vy(ii));     k2 = h * fVx(t(ii) + 0.5*h, vx(ii) + 0.5*k1, vy(ii) + 0.5*l1);     l2 = h * fVy(t(ii) + 0.5*h, vx(ii) + 0.5*k1, vy(ii) + 0.5*l1);     k3 = h * fVx(t(ii) + 0.5*h, vx(ii) + 0.5*k2, vy(ii) + 0.5*l2);     l3 = h * fVy(t(ii) + 0.5*h, vx(ii) + 0.5*k2, vy(ii) + 0.5*l2);     k4 = h * fVx(t(ii) + h, vx(ii) + k3, vy(ii) + l3);     l4 = h * fVy(t(ii) + h, vx(ii) + k3, vy(ii) + l3);     vx(ii+1) = vx(ii) + (1/6)*(k1 + 2*k2 + 2*k3 + k4);     vy(ii+1) = vy(ii) + (1/6)*(l1 + 2*l2 + 2*l3 + l4); end  % Plot the solution figure(1) plot(t, vx, t,vy) xlabel('Time (s)') ylabel('Velocities (m/s)') legend('Vx', 'Vy') ‘ld: warning: directory not found for option’