Hamilton's equations, phase space

Equipment

• Phase-plane canvas $(q,p)$ showing the oscillator orbit
• Solid (q,p) and dashed transformed (Q,P) curves drawn together
• Slider “Năng lượng E” (energy) and “Tạm dừng” button

Procedure

1. Read a phase orbit

   Let the oscillator run and watch the point trace a closed loop in $(q,p)$. Each point is a state, not a physical position; for $H=p^2/2m+\tfrac12m\omega^2q^2$ the loop is an ellipse enclosing area $2\pi E/\omega$.

2. Check the flow direction

   Pause at the top of the loop where $\dot q=\partial H/\partial p=p/m$ vanishes and $\dot p=-\partial H/\partial q=-m\omega^2q$ is at its extreme — the orbit turns there. Check the motion is clockwise and consistent with $\dot q$ and $\dot p$ signs at each quadrant.

3. Compare the transformed coordinates

   Look at the dashed curve $Q=q$, $P=p+\alpha q$ ($\alpha=0.7$): the ellipse is sheared and tilted, yet the enclosed area is unchanged since $\mathrm dQ\wedge\mathrm dP=\mathrm dq\wedge\mathrm dp$. This is a canonical transformation — same dynamics, new coordinates.

4. Scale the energy

   Move the energy slider E and watch the ellipse grow: its semi-axes scale as $\sqrt{E}$ while the area grows linearly. Predict the orbit's shape at double E, then verify — and explain why phase-space area conservation is called Liouville's theorem.

Simulation

Experiment history

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