2.1.9
Intuition Theorem ki ek-line wali roohHar continuous symmetry of the action ek conserved quantity deti hai.
Agar physics mein kuch badalta nahi jab tum shift/rotate/wait karo, toh kuch constant rehta hai.
Time mein shift → koi change nahi → energy conserved.Space mein shift → koi change nahi → momentum conserved.Space mein rotate → koi change nahi → angular momentum conserved.
YEH almost magical kyun lagta hai par actually obvious hai? Ek symmetry ka matlab hai ki Lagrangian kisi coordinate ki "parwah nahi karta." Agar L
Definition Action, Lagrangian, Euler–LagrangeAction hai S = ∫ t 1 t 2 L ( q i , q ˙ i , t ) d t S extremize karte hain (δ S = 0 Euler–Lagrange equation milti hai:
d t d ( ∂ q ˙ i ∂ L ) − ∂ q i ∂ L = 0 Conjugate (canonical) momentum hai p i ≡ ∂ q ˙ i ∂ L
YEH symmetry yahan kya hai? Coordinates (ya time) ki ek continuous transformation jo ek smooth parameter ϵ action invariant rehta hai (ya sirf ek boundary term se badalta hai).
Yeh ~80% exam problems handle karta hai aur sabse clean derivation hai.
Intuition Cyclic coordinates momentum kyun conserve karte hainEk coordinate q k cyclic (ignorable) hota hai agar L nahi karta: ∂ q k ∂ L = 0 d t d ∂ q ˙ k ∂ L − = 0 ∂ q k ∂ L = 0 ⇒ d t d p k = 0. p k "L q k q k L
Woh shift q k → q k + ϵ exactly ek continuous symmetry hai. Noether's theorem iska grown-up version hai jo tab bhi kaam karta hai jab L lekin action ki value invariant rahe.
Goal: Ek infinitesimal transformation of coordinates consider karo
q i → q i ′ ( ϵ ) = q i + ϵ δ q i , ϵ small , δ q i L invariant hai: δ L = 0 ϵ
Definition Noether's Theorem (point-symmetry form)Agar δ L = 0 d t d ∑ i p i δ q i = 0 Noether charge
Q = i ∑ ∂ q ˙ i ∂ L δ q i = i ∑ p i δ q i = const
L δ L = ϵ d t d F ∫ d t d F d t δ q Q = ∑ i p i δ q i − F = const . full Noether statement hai; δ L = 0 F = 0
Worked example (a) Time translation → EnergySymmetry: time t → t + ϵ L explicitly t t energy function ban jaata hai:
h = ∑ i q ˙ i ∂ q ˙ i ∂ L − L = ∑ i p i q ˙ i − L . δ q δ q i = q ˙ i δ L = d L / d t F = L h L = T − V T h = T + V = E Energy conserved hai kyunki laws time ke saath nahi badalte.
Worked example (b) Space translation → Linear momentumSymmetry: saare particles r a → r a + ϵ n ^ δ r a = n ^ Q = ∑ a p a ⋅ n ^ = n ^ ⋅ P total Yeh step kyun? Translation invariance ka matlab hai L relative positions par depend karta hai (koi absolute origin nahi), isliye woh center-of-mass coordinate par depend nahi kar sakta → woh coordinate cyclic hai → n ^
Worked example (c) Rotation → Angular momentumSymmetry: axis n ^ ϵ δ r a = n ^ × r a Q = ∑ a p a ⋅ ( n ^ × r a ) = n ^ ⋅ ∑ a ( r a × p a ) = n ^ ⋅ L total . Yeh step kyun? Humne scalar-triple-product identity p ⋅ ( n ^ × r ) = n ^ ⋅ ( r × p ) n ^
Worked example Plane par free particle — do symmetries check karoL = 2 1 m ( x ˙ 2 + y ˙ 2 ) Step 1 — x δ x = 1 , δ y = 0 δ L = 0 L x Q = p x = m x ˙ Kyun? x Step 2 — rotation: δ x = − y , δ y = x δ r = z ^ × r δ L = m ( x ˙ δ x ˙ + y ˙ δ y ˙ ) = m ( x ˙ ( − y ˙ ) + y ˙ x ˙ ) = 0 Q = p x δ x + p y δ y = m x ˙ ( − y ) + m y ˙ ( x ) = m ( x y ˙ − y x ˙ ) = L z Yeh step kyun? Yeh exactly angular momentum hai — confirm karta hai rotation ↔ L z
Common mistake Steel-manned common errorsMistake 1: "Energy hamesha conserved hoti hai."
Kyun sahi lagta hai: intro physics mein energy conservation ko universal hammer ki tarah padhaya jaata hai.
Fix: energy (Hamiltonian h L koi explicit time dependence nahi hoti. Ek driven/time-dependent potential time-translation symmetry tod deti hai → energy conserved nahi. Noether tumhe exactly batata hai kab.
Mistake 2: Canonical momentum = m v
Kyun sahi lagta hai: free particles mein p = m q ˙ Fix: conserved Noether charge p i = ∂ L / ∂ q ˙ i p x = m x ˙ + q A x
Mistake 3: Yeh sochna ki discrete symmetries (parity, reflection) Noether charges deti hain.
Kyun sahi lagta hai: woh phir bhi "symmetries" hain.
Fix: Noether ko ek continuous parameter ϵ d / d ϵ
Recall Feynman: ek 12-saal ke bacche ko explain karoSocho tum ek game khel rahe ho aur notice karte ho: koi farq nahi padta table par kahan se start karo, game same hi khelta hai. Yeh "koi farq nahi padta" ek hidden rule hai. Har baar jab game mein ek "doesn't-matter" rule hota hai, ek secret number hota hai jo khelne ke dauran kabhi nahi badalta. Agar game same dikhta hai chahe tum kahan khade ho → ek "push number" (momentum) frozen hai. Agar same dikhta hai chahe tum kis taraf munh karo → ek "spin number" (angular momentum) frozen hai. Agar same dikhta hai chahe tum kab khelo → ek "energy number" frozen hai. Noether ka bada idea: har "doesn't-matter" ke saath ek frozen number aata hai.
T ime, P osition, R otation symmetry → E nergy, M omentum, A ngular momentum conserved.
"Time-Position-Rotation gives Energy-Momentum-Angular."
Noether's theorem ek sentence mein Har continuous symmetry of the action ek conserved quantity se correspond karti hai.
Kaun si symmetry energy conserve karti hai? Time-translation invariance (L t
Kaun si symmetry linear momentum conserve karti hai? Spatial-translation invariance (koi absolute origin nahi).
Kaun si symmetry angular momentum conserve karti hai? Rotational invariance (space ki isotropy).
General Noether charge formula Q = ∑ i ∂ q ˙ i ∂ L δ q i − F δ L = d F / d t
Cyclic coordinate ki definition Woh coordinate jis par L
Cyclic coordinate p k EL deta hai p ˙ k = ∂ L / ∂ q k = 0
Kya energy hamesha conserved hoti hai? Nahi — sirf tab jab time-translation symmetry ho (L t
Kya discrete symmetries (parity) Noether charges deti hain? Nahi; Noether ko ek continuous parameter chahiye.
Infinitesimal rotation about n ^ δ r Angular momentum paane ke liye use ki gayi identity Conjugate momentum ki definition p i = ∂ L / ∂ q ˙ i
Lagrangian Mechanics — woh action principle jis par Noether's theorem based haiEuler-Lagrange Equation — derivation mein key stepConservation of Energy — time-translation caseConservation of Momentum — space-translation caseAngular Momentum — rotation caseHamiltonian Mechanics — energy function h Symmetry Groups & Lie Algebras — continuous symmetries Lie groups hainGauge Symmetry — local symmetries aur Noether's second theorem
Continuous symmetry of action
Action S = integral of L dt
