2.1.18
Jo system periodic motion karta hai (pendulum swing kare, planet orbit kare), uska phase space mein orbit ek closed loop hota hai (ya torus). Messy ( q , p )
Loop kitna bada hai? → yeh action J constant hai.Loop mein hum kahan hain? → yeh angle θ constant rate se wind karta hai: θ = ω t + θ 0
Toh integrable systems woh hain jinhe hum aisi coordinates mein transform kar sakte hain jahan sab kuch trivially ek seedhi line mein chalta hai . Mushkil hissa (frequencies dhundhna) sirf areas compute karne tak aa jaata hai.
Hamilton ke equations q ˙ = ∂ H / ∂ p p ˙ = − ∂ H / ∂ q canonical transformation ( q , p ) → ( θ , J )
Naya Hamiltonian sirf J H = H ( J )
Tab J ˙ = − ∂ H / ∂ θ = 0 θ ˙ = ∂ H / ∂ J = ω ( J ) ek number per degree of freedom mein squeeze kar deta hai — frequency ω bina poori time evolution solve kiye . Yahi 80/20 payoff hai.
Definition Action variableEk 1-DOF system ke liye jisme periodic motion hai, action woh phase-space area hai jo ek cycle se enclosed hoti hai, 2 π J ≡ 2 π 1 ∮ p d q ∮ ek complete period ke bounded motion par hai. J
Definition Angle variableIska canonical conjugate == θ == 2 π θ ( t ) = ω t + θ 0 , ω = ∂ J ∂ H .
Recall Self-derivation checkEk period mein ∮ d θ = 2 π θ = ∂ W / ∂ J ∮ d θ = dJ d ∮ ∂ q ∂ W d q = dJ d ∮ p d q = dJ d ( 2 π J ) = 2 π . J 2 π
H = 2 m p 2 + 2 1 m ω 0 2 q 2 = E Step A — p ( q , E ) p = ± 2 m E − m 2 ω 0 2 q 2 Kyun? Bas H = E p
Step B — phase orbit ek ellipse hai jiske semi-axes a = 2 E / ( m ω 0 2 ) q b = 2 m E p Kyun? 2 m E p 2 + 2 E / m ω 0 2 q 2 = 1
Step C — area / 2 π ∮ p d q = area = π ab = π m ω 0 2 2 E 2 m E = ω 0 2 π E . J = 2 π 1 ⋅ ω 0 2 π E = ω 0 E ⇒ E = ω 0 J . Ellipse area kyun use karo? Kyunki ∮ p d q wahi enclosed area hai — koi integration nahin chahiye.
Step D — frequency: ω = dJ d E = ω 0 .
Bonus (quantum link): E = ω 0 J J = ℏ ( n + 2 1 ) E n = ℏ ω 0 ( n + 2 1 )
0 L E = p 2 /2 m Step A: p = 2 m E Step B — ek period = right jaana aur wapas aana: ∮ p d q = p ⋅ L + p ⋅ L = 2 p L 2 L Step C: J = 2 π 1 2 p L = π p L = π L 2 m E E = 2 m L 2 π 2 J 2 Step D: ω = dJ d E = m L 2 π 2 J = m L 2 π 2 ⋅ π L 2 m E = L π v v = p / m T = 2 π / ω = 2 L / v
Definition Liouville integrabilityn integrable hai agar uske paas n F 1 , … , F n in involution hain: { F i , F j } = 0 i , j
Intuition Integrability se kya milta haiLiouville–Arnold theorem kehta hai: bounded level set { F i = c i } n T n ( θ i , J i ) J ˙ i = 0 , θ ˙ i = ω i ( J ) = ∂ J i ∂ H . ek doughnut par seedhi line mein winding n ω i
J p Kyun sahi lagta hai: J p q J = L Fix: J area-integral 2 π 1 ∮ p d q constant of motion averaged over a cycle , na ki instantaneous p
Common mistake Cycle ke there-and-back jaane ko bhool jaana.Kyun sahi lagta hai: p d q q min q ma x Fix: contour ∮ poora closed loop hai; libration ke liye q min → q ma x → q min q 2 π
Common mistake Yeh sochna kiH = H ( J ) Kyun sahi lagta hai: E J Fix: H = H ( J ) canonical transformation ke baad hota hai jab θ ( q , p ) H q
Common mistake Yeh assume karna ki har system integrable hai.Kyun sahi lagta hai: low-dimensional examples (SHO, Kepler) sab kaam karte hain. Fix: integrability ke liye n most systems (e.g., double pendulum, three-body) mein yeh nahin hote aur woh chaotic hote hain. Integrable systems rare, khoobsurat exceptions hain.
Mnemonic Recipe yaad karo"AREA → INVERT → DIFFERENTIATE"
Action = Area (/2π), invert karo E(J) paane ke liye, differentiate karo ω paane ke liye.
Aur contour ke liye: "Round trip = whole loop."
Recall Feynman: ek 12-saal ke bachche ko samjhaoSocho ek bachcha swing par hai. Swing hamesha wohi loop trace karta hai aage-peeche — out, back, out, back. Do baatein sab kuch describe karti hain: (1) swing kitna bada hai (yahi action hai — basically loop ka size, aur yeh same rehta hai), aur (2) bachcha abhi swing mein kahan hai (yahi angle hai, jo steadily aage badhta hai jaise clock ki sui). Jab loop ka size pata ho, toh kitni swings per minute hongi yeh sirf size se predict kar sakte hain — poori motion dekhni nahin padti. Scientists inhe "swing-jaisi" systems isliye pasand karte hain kyunki yahi akele hain jo hum fully predict kar sakte hain. Real cheezein (jaise ek chaotic spinning top) itni acchi nahin hoti.
Hamiltonian mechanics — action-angle ek special canonical transformation hai.Canonical transformations aur Generating functions — W ( q , J ) Hamilton–Jacobi equation — W Poisson brackets — involution { F i , F j } = 0 Liouville–Arnold theorem — phase space tori.Adiabatic invariants — slow parameter changes mein J Bohr–Sommerfeld quantization — ∮ p d q = 2 π ℏ ( n + 2 1 ) KAM theorem — perturbation ke under tori ka survival.
1-DOF periodic motion ke liye action variable define karo J = 2 π 1 ∮ p d q 2 π
Action-angle variables se oscillation frequency kaise nikaalte hain? ω = ∂ H / ∂ J = d E / dJ J ( E ) → E ( J )
Harmonic oscillator ke liye J E E = ω 0 J ω = d E / dJ = ω 0
Angle variable ek period mein kya karta hai? Yeh exactly 2 π θ = ω t + θ 0
J 1/2 π Taaki ∮ d θ = 2 π θ
Liouville integrability (n DOF) kya hai? n { F i , F j } = 0
Integrable system ke bounded motion ki geometry kya hai? Yeh ek n
J Yeh bhuul jaana ki libration ek round trip q min → q ma x → q min
Action-angle variables kaun sa generating function banata hai? Hamilton's characteristic function W ( q , J ) p = ∂ W / ∂ q θ = ∂ W / ∂ J
Action-angle old quantum theory se kaise connect karta hai? Bohr–Sommerfeld: J = ℏ ( n + 2 1 ) E n = ℏ ω 0 ( n + 2 1 )
Closed loop or torus in phase space
