![]() ![]() Taking the phase $a$ to be uniformly distributed basically says we are picking a point on the circle randomly. The angular coordinate is phase space is just the phase of the oscillator. ![]() So the radius of the circle is the energy. Then it is a circle.) The radius of the circle is $x^2 + p^2 = k x^2 + p^2/m = 2E$. (It will really be an ellipse, but lets choose units where $A=k=m=1$. You can convince yourself it is the circle. That being said we are ready to look at the trajectory of the oscillator in phase space. It's convential do view the phase space of a 1d oscillator as a plane with the $x$ axis being the position and the y axis being the momentum. The state of an oscillator is specified not just by the position of the oscillator but also by its momentum. Phase space is the space of states that the oscillator can be in. First let me explain phase space for completeness. Let's look at the trajectory of the oscillator in phase space. But I will try to answer your question anyway. Well you have detected a difference between a) and b) so you are doing better than me. ![]()
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