Action-angle variables — integrable systems: Visual walkthrough

2.1.18 · D2 · Physics › Analytical Mechanics › Action-angle variables — integrable systems

Step 1 — The playground: phase space

KYA kiya humne: "spring ko time mein hilte dekho" ko "ek dot ko $q$ – $p$ plane mein move hote dekho" se replace kiya. KYUN: kyunki bounded, repeating motion yahan ek closed loop draw karta hai — aur ek loop ka ek clean, measurable feature hota hai (uske andar ka area) jo ek wobbly time-graph mein nahi hota. PICTURE: dot (red) apne loop par baitha hai; arrows dikhate hain ki woh kis direction mein circulate karta hai.

Figure — Action-angle variables — integrable systems

Step 2 — Loop kyun close hota hai, aur "one period" ka matlab kya hai

$state ka function as total energy H (q, p) = ek fixed number, actual energy E$

KYA: loop ko naam diya — yeh level curve hai "energy = $E$ ". KYUN: kyunki dot constant energy ki curve nahi chhod sakta, wohi loop baar-baar trace hoti hai. One period = dot poora ek chakkar lagata hai aur apne start par wapas aata hai. PICTURE: teen energies $E_{1} < E_{2} < E_{3}$ ke liye teen nested loops. Badi energy ⇒ bada loop. Red loop woh hai jise hum track karte hain.

Step 3 — Action $J$ literally ek area hai

KYA: $J$ ko (loop area)/ $2 π$ define kiya. KYUN: ek area har loop ke liye, yani har energy ke liye ek fixed number hai — ideal conserved quantity. PICTURE: shaded loop; width $d q$ aur height $p$ ka red sliver dikhata hai ki $p d q$ kya measure karta hai, sab slivers sum karne se pehle.

Step 4 — Har tarah ke loop cover karo: libration vs rotation

Bounded motion ki do shapes hain, aur contour $\oint$ ka matlab dono ke liye alag hai:

Libration (aage-peechhe, jaise spring ya swing jo upar se nahi jaati): $q$ $q_{m i n}$ aur $q_{m a x}$ ke beech oscillate karta hai. Dot upar ( $p > 0$ ) se $q_{m i n} \to q_{m a x}$ jaata hai phir neeche ( $p < 0$ ) se $q_{m a x} \to q_{m i n}$ . Dono halves count hote hain — yahi woh famous factor of 2 hai (parent note mein box example).
Rotation (round and round, jaise pendulum jo upar se poora ghoom jaata hai): $q$ ek angle hai jo $2 π$ se badhta rehta hai aur wrap around ho jaata hai. Loop wapas aakar close nahi hota; angle wrap hone se close hota hai. Contour ek baar $0 \to 2 π$ chalti hai.

KYA: bounded motion ko do contour types mein split kiya. KYUN: kyunki $\oint p d q$ tab hi well-defined hai jab tumhe pata ho ki loop kaise close hota hai; yeh galat karne se $J$ 2 ke factor se ya poori shape se badal jaata hai. PICTURE: left = libration loop (khud par wapas aata hai, top red = forward, bottom = return); right = rotation "loop" (ek wavy band jo wrap karta hai kyunki $q$ periodic hai).

Step 5 — Harmonic oscillator ke liye area compute karo

Ellipse ka area $π ab$ hota hai (integration ki zaroorat nahi — geometry humein deti hai):

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

KYA: $H = E$ ko ek ellipse mein rearrange kiya aur uska area read kiya. Ellipse formula kyun use kiya? Kyunki $\oint p d q$ enclosed area hi hai — aur yahan enclosed shape ek perfect ellipse hai, jiska area ek known formula hai. Hum saara calculus skip karte hain. PICTURE: red ellipse jisme semi-axes $a$ aur $b$ labelled hain, area shaded.

Step 6 — AREA → INVERT: $E$ ko $J$ ke function ke roop mein pao

KYA: "area vs energy" ko "energy as a function of action", $E = E (J)$ mein baadla. INVERT kyun? Kyunki naya Hamiltonian action $J$ (hamara conserved coordinate) par depend karna chahiye, messy $(q, p)$ par nahi. $H = E (J)$ likhna exactly wahi hai. PICTURE: ek straight line — vertical axis par $E$ , horizontal axis par $J$ — slope $ω_{0}$ ke saath (red). Harmonic oscillator ke liye graph perfectly straight hai.

Step 7 — DIFFERENTIATE: frequency slope hai

Harmonic oscillator ke liye $E = ω_{0} J$ , toh slope constant hai: $ω = \frac{d E}{dJ} = ω_{0} . ✓$

Humne spring ki frequency $ω_{0}$ recover ki sirf ek loop ka area measure karke aur ek slope lekar — $\overset{q}{¨} = - ω_{0}^{2} q$ kabhi solve nahi kiya.

Recall

$J$ ki definition mein $2 π$ kyun hai? Angle partner $θ$ obey karta hai $θ = \partial W / \partial J$ . Ek poore trip mein, $\oint d θ = \frac{d}{dJ} \oint p d q = \frac{d}{dJ} (2 π J) = 2 π$ . Toh $\frac{1}{2 π}$ exactly wahi hai jo angle ko $0$ se $2 π$ tak ek baar ghoomata hai per period — ek clean clock hand.

KYA: $E (J)$ ko differentiate karke $ω$ paya. KYUN: kyunki naye coordinates mein $\dot{θ} = \partial H / \partial J$ constant hai, aur woh constant frequency hai — woh single number jo saari dynamics summarise karta hai. PICTURE: wohi $E$ – $J$ line jisme red mein slope triangle drawn hai; slope hi $ω$ hai.

Step 8 — Degenerate & edge cases (reader ko kabhi stranded mat chhhodo)

Intuition Kya hoga agar loop shrink ho ya graph bend ho?

$E \to 0$ (zero energy): ellipse origin par single point mein collapse ho jaati hai — koi motion nahi, $J = 0$ . Frequency formula phir bhi finite $ω_{0}$ deta hai kyunki yeh ek slope hai, aur line ka slope $J = 0$ par bhi survive karta hai. Ek frequency exist karta hai even infinitesimally small swing ke liye.
Non-linear system (jaise box, ya pendulum): $E$ – $J$ graph ek curve hai, line nahi. Tab $ω = d E / dJ$ energy ke saath badalta hai — bade swings ki alag frequencies hoti hain (yeh amplitude-dependent frequency hai). Particle in a box ke liye, $E = \frac{π ^{2} J ^{2}}{2 m L ^{2}}$ , ek parabola, toh $ω = \frac{π ^{2} J}{m L ^{2}}$ $J$ ke saath badhta hai.
Rotation contour (Step 4): wohi teen moves kaam karte hain, lekin $q$ ko there-and-back ki jagah ek baar $2 π$ se integrate karo — contour ko hamesha motion se match karo.

KYA: collapsed loop, bent graph, aur rotating contour check kiye. KYUN: recipe tumhe kabhi surprise nahi karni chahiye — har regime (zero, linear, non-linear, rotating) same teen moves se cover hota hai sahi contour ke saath. PICTURE: do $E$ – $J$ graphs side by side — ek straight line (harmonic, constant $ω$ ) vs ek curve (box, $ω$ growing); red slope triangle curve par do points par drawn hai yeh dikhane ke liye ki $ω$ badal raha hai.

Ek picture mein saari summary

Recipe yaad karne ke clozes:

J

words mein kya equals karta hai?

Ek loop ka enclosed phase-space area

2 π

se divided.

E

-vs-

J

ko frequency mein baadlne ke liye tum

differentiate karte ho (

ω = d E / dJ

Libration ke liye contour ek factor of

2 pick up karta hai (there-and-back).

Jab

E

–

J

graph curved hota hai, frequency

energy/amplitude par depend karti hai.

Recall Feynman retelling — poora walkthrough plain words mein

Ek bachche ko swing par imagine karo, lekin swing ko time mein dekhne ki jagah, ek dot draw karo: sideways hai kahan bachcha hai, upar-neeche hai kitni tez ja raha hai. Jab bachcha swing karta hai, woh dot wohi closed loop baar-baar trace karta hai. Ab sirf teen kaam karo. Ek: us loop ke andar ka area measure karo — yeh basically "swing ka size" hai, aur bachche ke swinging ke dauran fixed rehta hai; hum ise action kehte hain ( $2 π$ se shrink karne ke baad). Do: notice karo ki badi swing (zyaada energy) ka matlab bada loop hai, toh "energy vs loop-size" ka ek clean chart hai — use flip karo taaki loop-size se energy read kar sako. Teen: dekho ki woh chart kitna steep hai — woh steepness exactly hai kitni swings per minute bachcha karta hai. Aur yahan miracle hai: humne swings-per-minute sirf ek area measure karke aur ek slope read karke paaya. Humhe kabhi poori swing ko time mein tick karte nahi dekhna pada. Spring ke liye chart ek straight line hai, toh frequency kisi bhi swing size ke liye same hai; ek box mein bounce karte ball ke liye chart bend karta hai, toh bade bounces ka alag rhythm hota hai. Same teen moves dono mein: area, flip, slope.

Connections

Parent topic — woh recipe jo yeh page draw karta hai.
Hamiltonian mechanics — jahan se $(q, p)$ aur $H$ aate hain.
Canonical transformations · Generating functions · Hamilton–Jacobi equation — $W (q, J)$ ke peechhe ki machinery.
Liouville–Arnold theorem — multi-DOF torus generalisation.
Bohr–Sommerfeld quantization — jahan $J = ℏ (n + \frac{1}{2})$ Step 6 ko SHO spectrum mein baadl deta hai.
Adiabatic invariants · KAM theorem · Poisson brackets — integrability kya deti hai aur kahan toot-ti hai.