Question

Starting from the equation of motion of a
linear harmonic oscillator, show that its
phase space is an ellipse.

Answer 1

The equation of motion of a linear harmonic oscillator

$$\ddot{x}(t)+\omega^2x(t)=0$$

Solution of this equation defines the position of the oscillator with time

$$x(t)=A\sin(\omega t+\phi)$$

The linear momentum of a linear harmonic oscillator

$$p(t)=m\dot{x}(t)=m\omega A\cos(\omega t+\phi)$$

Using two last equations we obtain the phase trajectory equation

$$\frac{p^2}{(m\omega A)^2}+\frac{x^2}{A^2}=1$$

Therefore the phase space of a linear harmonic oscillator is an ellipse with semi-axes

$$a=m\omega A, \quad b=A$$