Why is the output of a series LC circuit sinusoidal?
Sinusoidal varying output for this circuit is something I can't
  understand


Consider a large flywheel connected to a non rotating motor via a rubber spring. Everything is at rest then suddenly the motor starts and rapidly reaches (say) 1200 rpm. The rubber spring starts twisting at the rate of 1200 rpm at the motor end but at the flywheel end it has barely moved because of the mass of the flywheel.

Eventually, the flywheel is seen to be turning slowly but gradually, it accelerates and some time later it reaches a speed of 1200 rpm.

What happens next is important. At the instant it reaches 1200 rpm there are many, many twists in the rubber spring and this represents stored energy but motor and flywheel are at the same speed. The rubber spring has no option; it exerts a rotating force on the only thing that can has freedom of movement (the flywheel) and, the flywheel ramps up in speed past 1200 rpm continuing until the rubber spring has exhausted its stored energy.

However, for each turn undone on the spring, the force transmitted by the spring's stored energy gets less and less until finally, when the flywheel has reached 2400 rpm, the spring is undone.

It doesn't stop there because the "problem" the spring now has is that the flywheel is rotating at 2400 rpm whilst the motor is stuck at 1200 rpm. So the rubber spring starts coiling up in the opposite direction, taking energy from the flywheel and slowing it down. Eventually both motor and flywheel are at the same speed but the inductor (ahem spring) has still got stored energy and this stored energy seeks to slow down the capacitor (ahem flywheel).

Eventually all the inductor/spring energy is spent and the capacitorflywheel is at 0V/0 rpm. The process then repeats and it's a sine wave of course.

Losses play a big part and the spring isn't perfect - it gets hot and this saps energy and the flywheel bearings get warm also. So it's a decaying sine wave and eventually the flywheel is continuously rotating at 1200 rpm.

It might help if you examined the voltage peaks you are seeing. I've altered your graph to put on 24V, 12V, 0V and -12V: -



If you analysed the period of oscillation you'd probably settle on something around 70 us. If you then used the LC formula for a tuned circuit i.e. this one: -

F = \$\dfrac{1}{2\pi\sqrt{LC}}\$ 

Then, if you plugged in the values for L and C you'd get a frequency of 15.915 kHz and that frequency has a period of 62.8 us. Perfectly normal behaviour for this "perfect" circuit.