2.1.16 · D1Canonical transformations — generating functions
Ek canonical transformation bas "sab possible states ke space" ke har point ko naya label dena hai — taaki future predict karne wali machine (Hamilton's equations) bilkul usi tarah kaam kare. Ek akela scalar recipe — generating function — batata hai kaise relabel karo bina un rules ko todte.
Parent note padhne se pehle, tumhe un shabdon mein fluent hona chahiye jo wahan use hote hain: phase space , coordinate , momentum , Hamiltonian , partial derivative , total time derivative , variation , new variables ( Q , P ) , generating function , Legendre transform , aur Poisson bracket . Yeh page inhe ek-ek karke scratch se build karta hai, ek aisi order mein jahan har symbol kamate hue aata hai use kiye jaane se pehle.
Definition Generalized coordinateq q q 1 , q 2 , … ek system kahan hai, ek instant mein, completely pin down karta hai . Wire par ek bead ke liye yeh wire ke saath distance hai; pendulum ke liye yeh swing ka angle hai.
Ek single number line imagine karo. Uspar q hi poora configuration hai.
Intuition "Generalized" kyun?Yeh word ka matlab hai ki hum x , y , z angle hai, length nahi. Hum jo bhi number sabse convenient ho choose karte hain — wahi freedom is topic ka poora point hai.
Definition Conjugate momentump p system kaise change ho raha hai — loosely, q p = m q ˙
Yahan q ˙ q
Definition Dot = time mein change ka rateq ˙ d t d q q q q ˙ q ¨
Intuition Momentum ke units (yeh ek baar karo, hamesha yaad rakho)Agar q m p = m q ˙ [ p ] = kg ⋅ s m = kg ⋅ m ⋅ s − 1 . alag units rakhte hain — yeh fact usi moment mein matter karta hai jab tum q p
Position akele future predict nahi kar sakti: neeche wala pendulum left, right move kar raha ho sakta hai, ya rest kar raha ho sakta hai. Tumhe dono chahiye — kahan hai aur kaise move kar raha hai. To inhe saath plot karo.
Intuition Upar ki figure mein kya notice karna haiViolet oval ek system ki poori history hai — ek closed loop matlab motion repeat hota hai. Akela magenta dot state hai abhi : iska horizontal position q p Orange arrow dikhata hai ki dot loop par creep karta hai jaise time ticks karta hai. Picture ko paRho jaise "future is a walk along this curve."
Ek plane (ya higher-dimensional space) jiska horizontal axis q p Ek dot = system ki ek complete state abhi. Jaise time guzarta hai, dot ek curve trace karta hai — system ki history.
Intuition Isko kyun chahiyeNewton ko x v ( q , p ) naye axes par le jaata hai — dots ka relabelling, physics ka change nahi. Un naye axes ke apne naam hain, jo hum aage introduce karte hain.
Definition New canonical variablesQ , P Q i = Q i ( q , p , t ) P i = P i ( q , p , t ) brand-new coordinates aur momenta hain — har ek purane ( q , p ) t usi phase-space ke dots ko alag numbers se label karte hain, bilkul jaise ek purane map par ek naya grid bichhana.
Intuition Capitals kyun, aur kyun bother karna?Capital letters bas ek bookkeeping habit hai: lower-case = purane labels, upper-case = naye labels. Hum bother karte hain kyunki ek clever regrid motion ko boring bana sakta hai — ek tangled curve ki jagah ek seedhi line. Parent note ka poora payoff (oscillator ko bina differential equation ke solve karna) ( Q , P )
Q P Kyun sahi lagta hai: letters q , p Fix: ek transformation ke baad, Q angle ho sakta hai aur P action (energy over frequency). "Coordinate" aur "momentum" words sirf is sense mein bachte hain ki woh wahi roles play karte hain jo q p
H ( q , p , t ) ( q , p ) total energy (kinetic + potential) q p H = 2 m p 2 + 2 1 m ω 2 q 2
Intuition Picture: phase space par ek height mapPhase space ko ek floor imagine karo aur H height — hills aur valleys. State-dot is landscape par slide karta hai, aur H naya landscape milta hai, traditionally K ( Q , P , t )
H q p t sirf p H
Definition Partial derivative∂ p ∂ H H p jabki q t . Curly ∂ d
Intuition Upar ki figure mein kya notice karna haiViolet bowl energy surface H q , p Magenta curve woh hai jo tumhe milta hai jab tum bowl ko fixed q ki line ke saath slice karte ho aur sirf p single curve ki steepness ∂ H / ∂ p
Intuition Partial kyun aur ordinaryd kis axis ke saath . ∂ / ∂ q ∂ / ∂ p q ˙ = ∂ H / ∂ p , p ˙ = − ∂ H / ∂ q
Parent topic ka poora quest yeh hai: ( q , p ) → ( Q , P ) K Q ˙ = ∂ P ∂ K , P ˙ = − ∂ Q ∂ K . canonical kahlata hai. Poori derivation ke liye Hamilton's equations dekho.
Yeh woh object hai jiske aaspaas poora parent note ghoomta hai, saaf taur par stated taaki baad mein kuch bhi surprise na kare.
Definition Generating functionF single scalar function — ek recipe — jisse poora transformation ( q , p ) → ( Q , P ) F derive karte ho.
Intuition Chaar "types" ka matlab (numbering scheme)F ek old aur ek new variable par depend karna chahiye per degree of freedom. Us pair ko choose karne ke exactly char tarike hain, isliye char types hain:
Type 1: F 1 ( q , Q ) Type 2: F 2 ( q , P ) Type 3: F 3 ( p , Q ) Type 4: F 4 ( p , P )
F F 1 q Q
Master relation mein d t d F
Definition Total vs partial time derivativeAgar F = F 1 ( q , Q , t ) q , Q khud time ke saath change hote hain, to
d t d F = ∂ q ∂ F q ˙ + ∂ Q ∂ F Q ˙ + ∂ t ∂ F . Total d / d t sab tarike count karta hai — directly t Partial ∂ F / ∂ t
d F / d t ∂ F / ∂ t q Q K = H + ∂ F / ∂ t partial use karta hai; master relation total use karta hai. Inhe straight rakhna hi poora game hai.
Definition Variation symbolδ δ δ ∫ ( … ) d t = 0 stationary banati hai (saari nearby paths ke beech ek valley ya ridge).
Intuition Upar ki figure mein kya notice karna haiNavy line ek pinned start dot se ek pinned end dot tak sachi path hai. Dashed magenta/orange/violet curves nearby "imagined" paths hain jo usi do endpoints share karti hain. Kyunki yeh sab ek saath start aur end karti hain, ∫ d t d F d t = F ( end ) − F ( start ) identical hai — isliye yeh affect nahi kar sakti ki kaunsi path jeetegi.
Intuition Endpoints kyun pinned hone chahiye (isliyeF d F / d t ∫ d t d F d t = F ( end ) − F ( start ) zero hai. Isliye ek generating function legal hai — Section 8 ki master relation poori tarah is fact par tikhi hai.
Type-1 recipe F 1 ( q , Q ) F 2 ( q , P ) new coordinate Q P paRega. Woh trade ek Legendre transform hai, aur ise step by step dekhna worthwhile hai.
Definition Legendre swap — step by stepGoal: ek variable Q P = ∂ F / ∂ Q
Shuru karo differentials ke liye product rule se: d ( P Q ) = P d Q + Q d P (Yeh bas ek product ki derivative hai, d Rearrange karo d Q P d Q = d ( P Q ) − Q d P PaRho: left side d Q d P d ( P Q ) d Q "d P ho gayi.Absorb karo boundary piece d ( P Q ) F 2 = F 1 + P Q Q P
Yahi poora swap hai: ek variable (Q P P Q
Intuition Yahi exact trick is topic mein kyun appear hoti haiF 1 ( q , Q ) ( q , P ) Legendre transformation dekho; yahi trick Lagrangian ko Hamiltonian mein bhi badalta hai.
Definition Poisson bracket (one degree of freedom)Phase space ke do functions f , g { f , g } = ∂ q ∂ f ∂ p ∂ g − ∂ p ∂ f ∂ q ∂ g . ki do quantities phase space mein kitni intertwine hain .
Intuition Check ke roop mein yeh kyun matter karta haiGenerating-function machinery automatically yeh brackets guarantee karti hai, lekin tum ek raw map ( q , p ) → ( Q , P ) Poisson brackets mein.
Master relation with dF over dt
Canonical transformations
Yeh seedha Symplectic geometry , Liouville's theorem , Hamilton–Jacobi equation aur Action–angle variables mein flow karta hai jab parent topic master ho jaata hai.
q ˙ q d q / d t
Agar q p = m q ˙ Kilogram-metres per second, kg ⋅ m ⋅ s − 1 q
Phase space mein ek single point kya represent karta hai? System ki ek complete state: kahan hai (q p
Q P New coordinates aur momenta, har ek purane ( q , p , t )
Zyaadatar problems mein H Total energy (kinetic + potential) q p
∂ H / ∂ p ∂ d Kyunki H ∂ p q t
Hamilton's do equations bolo. q ˙ = ∂ H / ∂ p p ˙ = − ∂ H / ∂ q
Ek sentence mein generating function F Ek single scalar recipe jisse differentiation karke poora transformation ( q , p ) → ( Q , P )
Woh master relation likho jo sab canonical transformations follow karte hain. ∑ p i q ˙ i − H = ∑ P i Q ˙ i − K + d t d F
Naya Hamiltonian K H K = H + ∂ F / ∂ t F
Chaar generator types kis cheez par keyed hain? F F 1 ( q , Q ) , F 2 ( q , P ) , F 3 ( p , Q ) , F 4 ( p , P )
d F / d t ∂ F / ∂ t Total d F / d t q , Q ∂ F / ∂ t t
Generator types ke beech Legendre swap kaunsi algebraic identity power karta hai? P d Q = d ( P Q ) − Q d P d Q d P
Multi-dimensional Poisson bracket { f , g } ∑ k ( ∂ q k ∂ f ∂ p k ∂ g − ∂ p k ∂ f ∂ q k ∂ g )
Kaun se fundamental brackets certify karte hain ki ( q , p ) → ( Q , P ) { Q i , P j } = δ ij { Q i , Q j } = { P i , P j } = 0
