Action-angle variables — integrable systems: Visual walkthrough

Step 1 — The playground: phase space

WHAT we did: replaced "watch the spring wobble in time" with "watch a dot move in a $q$ – $p$ plane." WHY: because bounded, repeating motion draws a closed loop here — and a loop has a clean, measurable feature (its enclosed area) that a wobbly time-graph does not. PICTURE: the dot (red) sits on its loop; the arrows show which way it circulates.

Figure — Action-angle variables — integrable systems

Step 2 — Why the loop closes, and what "one period" means

$total energy as a function of state H (q, p) = a fixed number, the actual energy E$

WHAT: we named the loop — it is the level curve "energy = $E$ ". WHY: because the dot cannot leave a curve of constant energy, the same loop is retraced again and again. One period = the dot goes all the way around once and returns to its start. PICTURE: three nested loops for three energies $E_{1} < E_{2} < E_{3}$ . Bigger energy ⇒ bigger loop. The red loop is the one we track.

Step 3 — The action $J$ is literally an area

WHAT: defined $J$ as (loop area)/ $2 π$ . WHY: an area is one fixed number per loop, i.e. per energy — the ideal conserved quantity. PICTURE: the loop shaded; the red sliver of width $d q$ and height $p$ shows what $p d q$ measures before we sum all slivers.

Step 4 — Cover every kind of loop: libration vs rotation

There are two shapes of bounded motion, and the contour $\oint$ means something different for each:

Libration (back-and-forth, like a spring or a swing that doesn't go over the top): $q$ oscillates between $q_{m i n}$ and $q_{m a x}$ . The dot goes $q_{m i n} \to q_{m a x}$ along the top ( $p > 0$ ) then $q_{m a x} \to q_{m i n}$ along the bottom ( $p < 0$ ). Both halves count — this is where the famous factor of 2 comes from (the box example in the parent note).
Rotation (round and round, like a pendulum swinging fully over the top): $q$ is an angle that keeps increasing through $2 π$ and wraps around. The loop does not close by coming back; it closes by the angle wrapping. The contour runs $0 \to 2 π$ once.

WHAT: split bounded motion into two contour types. WHY: because $\oint p d q$ is only well-defined once you know which way the loop closes; getting this wrong changes $J$ by a factor of 2 or its whole shape. PICTURE: left = libration loop (returns on itself, top red = forward, bottom = return); right = rotation "loop" (a wavy band that wraps because $q$ is periodic).

Step 5 — Compute the area for the harmonic oscillator

The area of an ellipse is $π ab$ (no integration needed — geometry hands it to us):

$\oint p d q = π ab = π \frac{2 E}{m ω _{0}^{2}} 2 m E = \frac{2 π E}{ω _{0}}$

WHAT: rearranged $H = E$ into an ellipse and read off its area. WHY use the ellipse formula? Because $\oint p d q$ is the enclosed area — and the enclosed shape here is a perfect ellipse, whose area is a known formula. We skip all calculus. PICTURE: the red ellipse with semi-axes $a$ and $b$ labelled, area shaded.

Step 6 — AREA → INVERT: get $E$ as a function of $J$

WHAT: turned "area vs energy" into "energy as a function of the action", $E = E (J)$ . WHY invert? Because the new Hamiltonian must depend on the action $J$ (our conserved coordinate), not on the messy $(q, p)$ . Writing $H = E (J)$ is exactly that. PICTURE: a straight line — $E$ on the vertical axis, $J$ on the horizontal — with slope $ω_{0}$ (red). For the harmonic oscillator the graph is perfectly straight.

Step 7 — DIFFERENTIATE: the frequency is the slope

For the harmonic oscillator $E = ω_{0} J$ , so the slope is constant: $ω = \frac{d E}{dJ} = ω_{0} . ✓$

We recovered the spring's frequency $ω_{0}$ purely by measuring the area of a loop and taking a slope — never solving $\overset{q}{¨} = - ω_{0}^{2} q$ .

Recall Why

$2 π$ in the definition of $J$ ? The angle partner $θ$ obeys $θ = \partial W / \partial J$ . Over one full trip, $\oint d θ = \frac{d}{dJ} \oint p d q = \frac{d}{dJ} (2 π J) = 2 π$ . So the $\frac{1}{2 π}$ is exactly what makes the angle run once around from $0$ to $2 π$ per period — a clean clock hand.

WHAT: differentiated $E (J)$ to get $ω$ . WHY: because in the new coordinates $\dot{θ} = \partial H / \partial J$ is constant, and that constant is the frequency — the single number that summarises all the dynamics. PICTURE: the same $E$ – $J$ line with the slope triangle drawn in red; the slope is $ω$ .

Step 8 — Degenerate & edge cases (never leave the reader stranded)

WHAT: checked the collapsed loop, the bent graph, and the rotating contour. WHY: the recipe must never surprise you — every regime (zero, linear, non-linear, rotating) is covered by the same three moves with the right contour. PICTURE: two $E$ – $J$ graphs side by side — a straight line (harmonic, constant $ω$ ) vs a curve (box, $ω$ growing); the red slope triangle is drawn at two points on the curve to show $ω$ changing.

The one-picture summary

Recall-the-recipe clozes:

J

equals what, in words?

The enclosed phase-space area of one loop divided by

2 π

To turn

E

-vs-

J

into the frequency you

differentiate (

ω = d E / dJ

For libration the contour picks up a factor of

2 (there-and-back).

When the

E

–

J

graph is curved, the frequency

depends on energy/amplitude.

Recall Feynman retelling — the whole walkthrough in plain words

Picture a kid on a swing, but instead of watching the swing, draw a dot: sideways is where the kid is, up-and-down is how fast they're going. As the kid swings, that dot traces the same closed loop over and over. Now do just three things. One: measure the area inside that loop — that's basically the "size of the swing", and it stays fixed while the kid swings; we call it the action (after shrinking it by $2 π$ ). Two: notice that a bigger swing (more energy) means a bigger loop, so there's a clean chart of "energy vs loop-size" — flip it around to read energy off the loop-size. Three: look at how steep that chart is — that steepness is exactly how many swings per minute the kid makes. And here's the miracle: we found the swings-per-minute just from measuring an area and reading a slope. We never had to sit and watch the whole swing tick through time. For a spring the chart is a straight line, so the frequency is the same for any swing size; for a ball bouncing in a box the chart bends, so bigger bounces have a different rhythm. Same three moves either way: area, flip, slope.

Connections

Parent topic — the recipe this page draws.
Hamiltonian mechanics — where $(q, p)$ and $H$ come from.
Canonical transformations · Generating functions · Hamilton–Jacobi equation — the machinery behind $W (q, J)$ .
Liouville–Arnold theorem — the multi-DOF torus generalisation.
Bohr–Sommerfeld quantization — where $J = ℏ (n + \frac{1}{2})$ turns Step 6 into the SHO spectrum.
Adiabatic invariants · KAM theorem · Poisson brackets — what integrability buys and where it breaks.