coupled-oscillatorsharmonic-oscillatoroscillatorsterminology
Coupling, according to wiki, is the condition of two systems when they interact with each other.
Now, I came across the terms harmonic oscillator and coupled oscillators. Now,what is the difference between the two? I read the wiki article of oscillation; but in vain. They only wrote that the later has higher degrees of freedom. What is meant by that?
1) Yes, you are right, but it really does not make much difference: sine and cosine are pretty much the same thing, with one just shifted over by a phase from the other.
2) They are assuming that the two frequencies are close, so $\overline\omega$ is also close to them and somewhere in between, but $\delta$ as the difference of two almost equal quantities is much smaller. So you hear a wave of frequency $\overline\omega$ but the wave is not of constant amplitude: it is "modulated" by the $\cos \delta t$ factor and you hear a (relatively) slow variation in loudness. If you take two tuning forks with identical frequency and wrap a rubber band around the leg of one to change its frequency just a little bit, you can start them vibrating and hold them close to your ear. You can then hear the wave at the basic frequency but the loudness will go up and down relatively slowly (the phenomenon is called beats and was used when making tuning forks: one was a standard and the other was modified until there were no beats produced. It's also used in tuning musical instruments: when there are no beats (i.e. $\delta = 0$), the two instruments are exactly in tune).
So you have a slow oscillation superimposed on a fast one:
(I got the picture from the above article in Wikipedia, so the notation is not the same, but it should be clear enough: what they call $\pi f_S$ is what you call $\delta$ and what they call $2\pi f_R$ is what you call $\overline\omega$).
3) Note then when $\cos \delta t$ is 0 then $\sin \delta t$ is maximum (or minimum) and vice versa. So at $t=0$ the amplitude of the first pendulum is maximum and the amplitude of the second is 0 (because at $t=0$, $\cos \delta t = 1$, $\sin \delta t = 0$). But some time later, when $\delta t = \frac{\pi}{2}$, the cosine is 0 and the sine is maximum (in absolute value). At that time, the amplitude of the first pendulum is 0 and the amplitude of the second is maximum. In between, one amplitude increases while the other decreases and the energy sloshes back and forth between the two pendulums (pendula?). That's not so easy to see with tuning forks but it is easy to see with a slinky to provide the weak coupling and a couple of cans of soup suspended with strings (see Crawford's "Waves" in the Berkeley Physics series from the 1960s: he includes a bunch of fascinating "home experiments" in the book). There are YouTube videos that illustrate this if you don't want to set up your own experiment.
And you were right about $x$.
Best Answer
Consider the case that you have two oscillators (harmonic or otherwise). It is the case that either
If (1), the differential equations for the system are just two single oscillator equations, e.g., the acceleration of one mass does not depend on the position of the other mass.
If (2), the differential equations are 'mixed', e.g., the acceleration of either mass depends on the difference in the position of the masses.
The term harmonic oscillator is opposed to anharmonic oscillator.
The natural (unforced) oscillation of a harmonic oscillator is a sinusoid, i.e., has a single frequency.