Generalized momentum and the Hamiltonian

Equipment

• «1. Mặt H(q, p)» canvas: the oscillator's 3D energy surface, rotatable by dragging or arrow keys
• «2. Quỹ đạo trên mặt phẳng pha» canvas: the phase ellipse for one energy level
• «Tần số góc ω» (angular frequency) slider, 50–200 rad/s
• «Năng lượng E» (energy) slider, 20–180 J
• Readouts «H(q,p)=…, ω=…, E=…» and «qmax = √(2E/mω²) = …; pmax = √(2mE) = …»

Procedure

1. Read the Hamiltonian on the energy surface

   In section 1, move «Tần số góc ω» and «Năng lượng E» and read «H(q,p)=p²/(2m)+mω²q²/2; m=1 kg, ω=…, E=…». Rotate the surface by dragging to see its curvature grow with ω: the potential $m\omega^2q^2/2$ steepens, and planes H = const cut the surface along ellipses.

2. Measure the phase-ellipse semi-axes

   Go to section 2 and read «qmax = √(2E/mω²)» and «pmax = √(2mE)». Check yourself: at ω = 100 rad/s, E = 80 J (m = 1 kg) you should get $q_{max} = \sqrt{2\cdot80/100^2} = 0.126$ m and $p_{max} = \sqrt{160} = 12.65$ kg·m/s. Compare with the ellipse extents on the canvas and record.

3. Check the tangent flow of Hamilton's equations

   Watch the phase ellipse: at q = qmax, p = 0 and $\dot p = -m\omega^2 q < 0$, so the state point moves downward; at p = pmax, $\dot q = p/m > 0$, so it moves right — the orbit circulates clockwise in the (q, p) plane. Change ω and E: the ellipse rescales but the flow direction and $H = E$ are preserved.

4. Predict the effect of ω and E

   Predict before sliding: doubling ω halves qmax but leaves pmax unchanged (it depends only on E); raising E scales both semi-axes by $\sqrt{E}$. The ellipse area $A = \pi q_{max}p_{max} = 2\pi E/\omega$ — check by holding E, changing ω, and re-reading the two values: the product qmax·pmax must shrink exactly as 1/ω.

Simulation

Experiment history

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