Question

Starting with the equation of motion, prove that the phase space of a linear harmonic oscillator is elliptical.

Answer 1

The equation of motion of a harmonic oscillator has the following form:

$$\ddot{x} + \omega_0^2 x = 0$$

By introducing the new variable

$$y = \dot{x}$$

we transform the initial 2nd order ODE into the set of the two 1st order ODEs:

$$\dot{y} = - \omega_0^2 x \\ \dot{x} = y$$

Excluding time, we come to the single ODE defining the phase portrait of the system:

$$\frac{dy}{dx} = - \frac{\omega_0^2 x}{y}$$

Performing the integration, we obtain:

$$\frac{y^2}{2} + \frac{\omega_0^2 x^2}{2} = c^2$$

Introducing the notations

$$a^2 \equiv \frac{2 c^2}{\omega_0^2}, \quad b^2 \equiv 2 c^2,$$

we obtain

$$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$

which is the canonical equation of an ellipse.