2.1.18 · D1Action-angle variables — integrable systems
Jo system apni motion repeat karta hai, woh ek abstract state-plane mein ek closed loop trace karta hai (jise hum section 2 mein phase space kahenge) — aur us loop ko sirf do honest numbers se describe kiya ja sakta hai: kitna bada hai (the action) aur hum us par kitna aage hain (the angle). Parent page par jo bhi machinery hai, woh sab ek messy repeating motion ko us clean loop mein convert karne ke liye hai.
Parent note padhne se pehle, tumhare paas uski vocabulary honi chahiye. Neeche, har symbol aur idea ko zero se build kiya gaya hai, ek aisi sequence mein jahan ladder ki har rung apne neeche wali rung par tikti hai.
q p == q == cheez kahan hai (ek angle, ek distance — har tarike se move karne ke liye ek number). == p == kitni motion carry kar raha hai — roughly mass times velocity, p = m v
Intuition Do numbers kyun, ek kyun nahin?Agar main tumhe sirf yeh batata hoon ki pendulum "bottom par hai", tum agla instant predict nahin kar sakte: ya toh woh tezi se swing kar raha hoga ya bilkul still baitha hoga . Tumhein dono chahiye — kahan hai (q aur kaise move kar raha hai (p ( q , p ) abhi poora future fix kar deta hai — woh pair hi complete "state" hai.
Figure 1 dekho: ek dot us plane par jiska horizontal address q p
Woh plane (ya higher space) jiske axes q p = move karta hai, ek curve draw karta hai.
Intuition Aise kyun draw karte hain?Kyunki ek repeating motion yahan ek closed loop draw karta hai. Aana-jaana-aana tumhe waapas same ( q , p )
Figure 2 mein swinging pendulum ka dot ek oval mein ghoomta rehta hai, baar baar, kabhi usse nikalke nahin jaata.
Definition Periodic (bounded) motionMotion jo ek limited region mein rehti hai aur ek fixed time T repeat hoti hai. Ek swing, ek planet, ek spring par mass.
Intuition "Repeat" ke do flavours
Libration — aage-peechhe jaise ek swing: q oval .Rotation — chakkar-pe-chakkar jaise ek spinning wheel: q 0 → 2 π → 4 π … state har turn mein repeat hoti hai.
Dono phase space mein ek closed loop dete hain; parent note ka "there-and-back" factor of 2 libration se aata hai, aur "q 2 π
Intuition Rotation ke liye wrap-around (q ≡ q + 2 π q angle hai, isliye q = 0 q = 2 π same physical place hain — inhe identify karte hain, likha jaata hai q ≡ q + 2 π 2 π q = 2 π 2 π
T ω T ek full loop ke liye seconds. == ω == radians of loop-progress per second :
ω = T 2 π .
2 π ek full loop = 2 π , bilkul ek clock hand ke ek full turn ki tarah. Toh ω
Angle measure karne ka ek tarika unit circle par arc length se . Ek full circle 2 π ≈ 6.283 π
Intuition Yahan radians kyun?Angle variable θ 2 π = 2 π
∫ f d q q d q f f q neeche ka area . Yahan f q f = p
Intuition Action ke liye integral kyun?Action ek area in phase space hai. Area = height p q d q hi hai ∫ p d q
∮ p d q ∮ p d q poore closed loop ke around , start tak waapas, add karo. Result hai loop ke andar ka area .
Intuition Chhota circle kyun matter karta haiAgar tum sirf q m i n q m a x aadha hi milega (top). Circle batata hai ki neeche se bhi waapas aana hai — toh dono halves mil ke poora enclosed area fence karte hain. Figure 3 exactly yeh area shade karta hai.
Definition Orientation / sign conventionDirection matter karta hai. Hum loop ko ( q , p ) counterclockwise traverse karte hain (woh direction jis mein physical motion actually jaati hai: top par p > 0 q p < 0 q J positive hota hai. Clockwise traverse karo toh same number minus sign ke saath milega; hum hamesha woh direction chunte hain jo J > 0
Common mistake Return trip drop karnaEk taraf integrate karna aur loop ke bottom ko bhool jaana answer adha kar deta hai. ∮ poora loop, hamesha .
J J ≡ 2 π 1 ∮ p d q . loop ka area lo, 2 π Yeh ek single number hai jo batata hai "yeh orbit kitna bada hai".
2 π 2 π J constant rehta hai jab system evolve karta hai — loop ka size nahin badalta jab tum uske around march karte ho — aur yahi reason hai ki yeh ek acha "coordinate jo kabhi move nahin karta" banata hai.
Definition Angle variableθ J loop ke around tum kitna aage ho , aur constant rate se advance karta hai:
θ ( t ) = ω t + θ 0 . == θ 0 == initial phase — loop par woh jagah jahan tum time t = 0 θ 2 π
Socho ek clock hand loop ke centre par pin kiya hua, steady speed ω θ θ 0 J ( θ , J ) ( q , p )
H ( q , p ) ( q , p ) total energy (kinetic + potential) return karti hai. Aisi motion ke liye jo time ke saath nahin badlti, H E
Intuition Yahan kyun matter karta haiEquation H ( q , p ) = E p p ( q , E ) Hamiltonian mechanics dekho.
∂ J ∂ H dJ d E derivative ek rate of change / slope hai: "agar main neeche wale variable ko thoda nudge karun, toh upar wala kitna move karta hai?" Curly ∂ sirf ise badlo"; straight d H E
Intuition Derivatives frequency kyun dete hainParent ka punchline hai ω = dJ d E energy-vs-loop-size graph ka slope hai. Steeper graph → faster oscillation. Isliye tum ω bina motion solve kiye paa sakte ho — sirf areas ka ek slope measure karo.
Definition Canonical transformationEk change of coordinates ( q , p ) → ( θ , J ) Canonical transformations dekho.
Definition Generating functionW ( q , J ) old position q new momentum J manufacture karta hai:
p = ∂ q ∂ W , θ = ∂ J ∂ W . W q p J θ Generating functions aur Hamilton–Jacobi equation dekho jahan se W
Intuition Yeh gadget kyun chahiyeHum sirf new coordinates declare nahin kar sakte; unhe mechanics respect karni chahiye. W ( q , p ) ( θ , J )
Definition Poisson bracket{ F , G } ek kitna change hota hai jab tum doosre ke along flow karte ho . Agar { F , G } = 0 Poisson brackets dekho.
Intuition Multi-DOF ke liye kyunn n { F i , F j } = 0 Liouville–Arnold theorem ).
Ek doughnut surface. n n straight-line winding ban jaati hai. Figure 4 do angles ke saath 2-torus dikhata hai.
Har degree of freedom ka apna loop hai (apna circle). Do circles combined = doughnut ki surface; n n θ i
position q and momentum p
periodic motion draws a closed loop
period T and frequency omega
closed loop integral of p dq
action J equals loop area over two pi
angle theta winds at rate omega
Hamiltonian H equals E is the loop
canonical transform via generating W
clean coordinates theta and J
Poisson brackets in involution
Action angle variables topic
Right side cover karo; kya tum parent note padhne se pehle har cheez ka jawaab de sakte ho?
Phase space mein ek single point kya represent karta hai? System ki ek complete state — ek position q p
Hume q p q Position akele agla instant predict nahin kar sakta; tumhein yeh bhi chahiye ki kitni tezi se / kis direction mein move kar raha hai. Pair poora future fix kar deta hai.
Periodic motion phase space mein kaunsa shape draw karti hai? Ek closed loop (libration ke liye oval, rotation ke liye full circle).
Libration aur rotation mein kya fark hai? Libration aage-peechhe hai (there-and-back); rotation chakkar-pe-chakkar hai jismein q 2 π
Rotation ke liye, loop close kyun hota hai jabki q Kyunki q q ≡ q + 2 π q = 2 π q = 0 2 π
∮ ∫ ∮ closed loop ke around poora sum karta hai (whole area); ∫
Loop kis direction mein traverse karte hain, aur kyun? ( q , p ) J
Ek line mein action J Loop ka enclosed phase-space area 2 π J = 2 π 1 ∮ p d q
Area ko 2 π Taaki uska partner angle θ 2 π
Angle variable θ θ 0 θ ω θ 0 t = 0
Derivative kya hai, plain words mein? Ek slope — jab tum neeche wale ko nudge karte ho toh upar wala quantity kitna badlta hai.
Energy aur action se frequency ω ω = d E / dJ
Loop kaunsa equation hai? H ( q , p ) = E
Generating function W ( q , J ) p = ∂ W / ∂ q θ = ∂ W / ∂ J
Multi-DOF system integrable kab hota hai, aur uski motion kis shape par rehti hai? Jab uske paas n { F i , F j } = 0 n
