Foundations — Noether's theorem — symmetry ↔ conservation law

2.1.9 · D1 · Physics › Analytical Mechanics › Noether's theorem — symmetry ↔ conservation law

Isse pehle ki hum Noether's theorem ko state bhi kar sakein, humein har woh symbol kamaana hoga jo woh use karta hai. Parent note mein $q_i$, ${\dot{q}}_i$, $t$, $L$, $S$, $d$, $\partial$, $p_i$, $\delta$, $\times$, aur $\int$ aise use kiye jaate hain jaise tumhe pehle se pata ho. Yeh page har ek ko bilkul zero se banata hai, uss order mein jis order mein woh ek doosre par depend karte hain. Koi bhi cheez use nahi hogi jab tak draw na ho jaaye.

1. Position aur coordinate $q$

Figure s01 neeche $q$ ko ek slider ki tarah dikhata hai jiska value wire par uski position hai.

Topic ko yeh kyun chahiye. Noether poochhta hai "jab main ek coordinate ko shift karta hoon toh physics ka kya hota hai?" Tum woh shift nahi kar sakte jo tum naam nahi de sakte. $q_i$ har us direction ka naam hai jis mein system move ho sakta hai — aur symmetries inhi naamon ki shifts hain. Generalized coordinates kahaan se aate hain, yeh dekhne ke liye Lagrangian Mechanics dekho.

2. Time $t$ — woh clock jiske against hum sab kuch measure karte hain

Picture yeh socho: ek film reel. Har frame par ek time $t$ stamped hai; coordinate $q$ woh hai jo frame mein dikhta hai. $t$ frame counter hai, $1,2,3,\dots$ chalta rehta hai chahe screen par kuch bhi ho.

Topic ko yeh kyun chahiye. Noether ke teen headline results mein se do time par depend karte hain. "Physics jaise clock ticki hai waise nahi badlti" yeh time-translation symmetry hai, aur yeh energy ko freeze karti hai. Isliye $t$, $L(q,\dot{q},t)$ mein decoration nahi hai — yeh ek genuine input hai jise hum baad mein $ϵ$ se shift karenge. Neeche sab kuch (velocity, derivative, integral) is clock ke against measured change hai.

3. Differential $d$ aur derivative $\frac{d}{dt}$

Picture: $q$ (vertical) ko $t$ (horizontal) ke against plot karo. $\frac{dq}{dt}$ ek point par graph ki steepness hai — steep matlab $q$ har tick mein fast change karta hai. Yahi honest meaning hai jis par hum Noether ki derivation mein har baar $\frac{d}{dt}$ aane par lean karenge.

$d$ ko baaki sab se pehle kyun. Velocity, action ke slices, aur poora "rate of change of momentum is zero" punchline sab $d$ se bane hain. Yeh $\dot{q}$ se pehle, $\int$ se pehle, $\partial$ se pehle define hona chahiye.

4. Velocity aur dot: $\dot{q}$

Yeh notation kyun aur $\frac{dq}{dt}$ bar bar kyun nahi? Kyunki hum inhe dozens baar likhenge. Dot ek shorthand hai jo time-derivatives itni baar likhne se paida hua ki Newton ne upar ek dot rakh diya. Iska matlab hamesha "clock $t$ ke har tick par" hai. Yeh sawaal ka jawaab deta hai: "slider abhi kis taraf, aur kitni tezi se khisal raha hai?"

Figure s01 mein wapas, $q$ woh jagah hai jahaan blue bead hai; orange arrow $\dot{q}$ dikhata hai kis taraf aur kitni jaldi woh move hone wala hai.

5. Function machine: Lagrangian $L$

$L(q,\dot{q},t)$ padho as: "mujhe batao tum kahaan ho, kaise move kar rahe ho, aur kya time hai; main ek number wapas deta hoon." Commas ingredients ko alag karti hain — aur teesra ingredient exactly Section 2 ka clock $t$ hai.

Topic ko yeh kyun chahiye. Noether ki duniya mein "symmetry" ka matlab hai recipe $L$ nahi badlti jab tum kuch shift karte ho. Isliye $L$ woh object hai jiske against hum symmetries test karte hain. Dekho Lagrangian Mechanics.

6. Partial derivative: $\frac{\partial L}{\partial q_i}$

$\partial$ kyun aur plain $d$ kyun nahi? Kyunki $L$ ke bahut saare inputs hain. Plain $\frac{dL}{dt}$ unhe sab ko ek saath change hone deta (har ek clock ke saath chalte hue). Partial ek knob ka effect isolate karta hai. Yeh poochhhta hai: "kya $L$ is particular coordinate ki parwah karta hai?" Agar $\frac{\partial L}{\partial q_k}=0$, toh jawaab na hai — $q_k$ cyclic hai, aur yeh ek symmetry hai.

Figure s02 $L$ ko uske inputs ke upar ek landscape ki tarah dikhata hai; partial ek chosen direction mein slope hai.

7. Conjugate momentum: $p_i\equiv \frac{\partial L}{\partial {\dot{q}}_i}$

Ise momentum kyun kaha jaata hai? Ek free particle ke liye $L=\frac{1}{2}m{\dot{x}}^2$, hum paate hain $p_x=\frac{\partial L}{\partial \dot{x}}=m\dot{x}$ — woh jaana-maana $mv$. Lekin generally yeh kuch aur bhi ho sakta hai (dekho Conservation of Momentum aur parent mein vector-potential warning). Baat yeh hai: $p_i$ jo bhi hai Noether use freeze karega.

8. Integral aur action: $S={\int }_{t_1}^{t_2}Ldt$

Figure s03 is sum ko $L$ ke time ke against graph ke neeche area ki tarah dikhata hai.

Topic ko yeh kyun chahiye. Symmetry ki sabse gehri form hai "action $S$ unchanged hai," chahe $L$ khud ek boundary term se change ho jaaye. Action woh ultimate scorecard hai jise nature extremize karta hai — woh object jise symmetries preserve karni chahiye. Dekho Euler-Lagrange Equation.

9. Variation symbol: $\delta$

$d$ ki jagah naya symbol kyun? $d$ track karta hai cheezein kaise badlti hain jaise actual path par time flow hota hai (clock $t$ ticking). $\delta$ track karta hai cheezein kaise badlti hain jab hum same instant par ek nearby imagined path par jump karte hain. Change ke do different kinds ko do different symbols chahiye, warna algebra jhooth ban jaata hai.

10. Vectors, cross product $\times$, aur triple product

Figure s04 $\hat{n}\times \vec{r}$ ko ek infinitesimal rotation ke chhote sideways nudge ki tarah dikhata hai.

Dot $\cdot$ dot product hai: $\vec{a}\cdot \vec{b}$ = (woh kitna same way point karte hain) — ek single number, parallel hone par sabse bada, perpendicular hone par zero.

11. Yeh sab Noether ko kaise feed karta hai — map padhna

Ab jab har symbol define ho gaya, yeh dependency map hai. Ise bottom-up ek recipe ki tarah padho: har box ko samajhne se pehle usmein feed hone wali har cheez chahiye. Ek path zor se trace karo — coordinate $q$ aur uski velocity $\dot{q}$ milke Lagrangian $L$ banate hain; $L$ se ek partial derivative momentum $p$ deta hai; momentum plus variation $\delta q$ Noether charge $Q$ banata hai; aur $Q$, time / space / rotation ke liye padha jaaye, teen conservation laws ban jaata hai. Har arrow ka matlab hai "tail samajhne se pehle head nahi."

Neeche teeno conserved quantities poore topic ka payoff hain.

Equipment checklist

Khud test karo — right side cover karo aur reveal karne se pehle zor se jawaab do.

Jab upar ki har line automatic ho jaaye, tab main derivation par jao, aur deeper layers ke liye Hamiltonian Mechanics, Conservation of Energy, aur Symmetry Groups & Lie Algebras dobara dekho.