Question

What is the shape of the phase space trajectory for a harmonic oscillator?

• A. A straight line
• B. A circle
• C. An ellipse
• D. A parabola

Hint:

Conserved energy \(E = p^2/2m + \tfrac12 m\omega^2 x^2\) is the equation of an ellipse in the \((x,p)\) plane.

Answer 1

The Correct Option is C

Step 1: The motion of a harmonic oscillator is $x(t) = A\cos\omega t$ and its momentum is $p(t) = -mA\omega\sin\omega t$. These parametric equations trace out the phase-space curve as time advances.

Step 2: Eliminate time using $\cos^2\omega t + \sin^2\omega t = 1$: \[\left(\frac{x}{A}\right)^2 + \left(\frac{p}{mA\omega}\right)^2 = 1.\]

Step 3: This is the canonical equation of an ellipse in the $(x,p)$ plane, with $x$-semi-axis $A$ and $p$-semi-axis $mA\omega$. Because these lengths differ (unless artificially rescaled), the closed curve is an ellipse.

Step 4: The closed loop reflects the periodic, energy-conserving nature of the oscillator: each cycle returns the system to the same point in phase space, tracing the ellipse once per period.\[\boxed{\text{An ellipse}}\]