Phase Space of an Oscillator
For a harmonic oscillator, p^2/(m^2 omega^2 A^2) + x^2/A^2 = 1, so constant-energy trajectories are ellipses and their orientation shows time flow.
Why this shows up in the exam
Reading phase-space diagrams · Comparing oscillator energies · Distinguishing periodic and nonperiodic motion
Learn the idea
An ideal oscillator traces a closed ellipse in position-momentum space, with energy setting its size. A phase-space point stores both where the oscillator is and how it moves; one cycle carries that point once around a closed curve.
🧠 Memory hook: One state is one point; one cycle is one loop.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- x²/A² + p²/(m² omega² A²) = 1 — constant-energy phase-space ellipse
- E = p²/(2m) + m omega² x²/2 — Hamiltonian determining the orbit
- dx/dt = p/m — horizontal direction of phase-space motion
How to approach it
- 1Read intercepts on both axes
- 2Relate intercepts to amplitude and maximum momentum
- 3Use the signs of dx/dt and dp/dt to set direction
Common slip-ups that cost marks
- •Assuming every ellipse is a circle
- •Ignoring axis scales
- •Reversing the arrow direction at positive momentum
🌟 That's the whole idea — you've got this. Try the practice set below; every question you attempt makes it stick a little harder.
Original chapter practice
Original questions for this chapter, not past-paper questions or an exact mapping to this individual concept.
A mass of 1 kg is attached to a spring of force constant 100 N/m. Find the angular frequency of small oscillations.
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