Foundations — Constraints using Lagrange multipliers

2.1.10 · D1 · Physics › Analytical Mechanics › Constraints using Lagrange multipliers

Yeh parent topic ka vocabulary page hai. Ise ek baar padho aur baad ki har equation seedhi English jaisi lagegi.

0. Woh scene jo hum describe kar rahe hain

Is chapter mein sab kuch ek aisi object ke baare mein hai jo kisi jagah rehne ke liye majboor hai: ek bead jo wire par piroyi hui hai, ek mass jo rope se latka hai, ek ball jo bina phisale roll kar rahi hai. Object khud se aazad ghoomna chahti hai (gravity khichti hai), lekin kuch cheez kuch motions ko mana karti hai. Hamara poora kaam hai ki dono — aazadi aur pabandi — ko symbols se describe karen.

Tasveer dekho: bead curve par atki hui hai. Woh wire ke saath saath slide kar sakti hai (woh arrow allowed hai) lekin wire se bahar kabhi nahi ja sakti (woh arrow mana hai). Yeh split — saath vs bahar — apne dimaag mein rakho. Neeche har symbol in do directions mein se kisi ek ke baare mein baat karne ka tool hai.

1. Coordinates $q_k$ — woh numbers jo batate hain object kahan hai

Tasveer: bead ko plane mein locate karne ke liye tum uski horizontal position $x$ aur vertical position $y$ de sakte ho. Yeh do numbers hain, toh $q_1=x$, $q_2=y$. Ya, kyunki woh circle par rehti hai, tum radius $r$ aur angle $\theta$ de sakte ho — woh polar coordinates jo parent note pasand karta hai. Kisi bhi tarah, ek coordinate ek dial hai jise tum ghuma ke object ko hilate ho.

Topic ko yeh kyun chahiye: parent note ki har equation "$k=1,\dots ,n$ ke liye" chalti hai. Woh $n$ hai kitne dials tumhare paas hain. Bina pehle coordinates ke tum constraint bhi state nahi kar sakte. Unhen choose karne ki poori kahani ke liye Generalized Coordinates and Degrees of Freedom dekho.

2. Constraint function $f(q,t)=0$ — deewar ko equation ke roop mein likhna

Tasveer: radius $R$ ke hoop par ek bead ke liye, niyam hai "centre se tumhari doori hamesha $R$ hai." Ek function ke roop mein jo zero hona chahiye: $f=r-R=0.$ Agar $r=R$, toh $f=0$ — allowed. Agar $r=1.2R$, toh $f=0.2R\ne 0$ — forbidden. Allowed points ka set ($f=0$) ek surface hai (yahan, circle). $f$ ko ek aisi height map sochho jiska "sea level" ($f=0$) bilkul deewar hai.

$t$ kyun? Kabhi kabhi deewar khud time mein chalti hai (ek hoop utha ek uthaya ja raha hai, ek rope reeled in ho rahi hai). $f$ ko time $t$ par depend karne dena unhe cases cover karta hai. Agar deewar fixed hai, toh $f$ mein $t$ aata hi nahi — lekin hum general rehne ke liye $t$ likhte hain.

3. Partial derivative $\frac{\partial f}{\partial q_k}$ — ek direction mein deewar ki dhaul

Yeh pehla sachchi naya tool hai, toh hum ruk ke ise achhe se samajhte hain.

Yeh tool kyun chahiye aur koi nahi: hum jaanna chahte hain "deewar se bahar" kaunsi direction hai. Ek akela ordinary derivative nahi bata sakta, kyunki "bahar" ka matlab ho sakta hai $x$ nudge karna, ya $y$ nudge karna, ya kuch mix. Partial derivative hamare ko har direction alag se test karne deta hai, phir jawab jodne deta hai.

Tasveer: height-map surface $f$ par khado. $\partial f/\partial q_k$ woh dhaul hai jo tum feel karte ho agar $q_k$ direction mein ek kadam lo. $f=r-R$ ke liye: $\frac{\partial f}{\partial r}=1,\frac{\partial f}{\partial \theta }=0.$ $r$ mein kadam lene se $f$ rate $1$ par badlata hai (tum seedha sea level se bahar chadh jaate ho). $\theta$ mein kadam lene se $f$ bilkul nahi badlata — tum deewar ke saath saath slide karte ho. Woh zero mathematics ka yeh kehna hai "allowed direction $\theta$ wali hai."

4. Gradient $\nabla f$ — seedha deewar se bahar ki taraf dikhata arrow

Tasveer: hoop par, $\nabla f=(\partial f/\partial r,\partial f/\partial \theta )=(1,0)$ — ek arrow jo radially bahar ki taraf point karta hai, circle ke bilkul perpendicular. Yahi woh direction hai jis mein ek deewar ki normal force push kar sakti hai. Yeh is poore topic ka raaz ka kaabza hai.

5. Virtual displacement $\delta q_k$ — ek khayal mein "allowed" nudge

Tasveer: bead ko pakdo aur use wire ke saath thoda hilao — kabhi wire se bahar nahi. Move ki woh chhoti si phuskari $\delta q$ hai. Yeh "virtual" hai kyunki tum actually time nahi badhne dete; tum bas poochh rahe ho "kya hoga agar main ise is taraf nudge karun — kya yeh allowed hai?"

Frozen time kyun? Real motion mein object ka hilna aur deewar ka shayad hilna mix ho jaata hai. Object ki abhi deewar par aazadi ko isolate karne ke liye, hum $t$ ko rok dete hain. Isliye constraint ki variation apna $\partial f/\partial t$ term drop kar deti hai: $\delta f={\sum }_k\frac{\partial f}{\partial q_k}\delta q_k=0.$

6. Kinetic aur potential energy $T$, $V$, aur Lagrangian $L=T-V$

Tasveer: bead ko hoop par adha upar imagine karo. Uski kuch speed hai (yeh $T$ hai) aur kuch height hai (yeh $V$ hai). $L$ unhe ek number mein blend karta hai jo, jab hum ise sahi tarike se differentiate karte hain, Newton ke laws bahar nikalte hain — lekin kisi bhi coordinates mein, chahe ajib kyun na hon. Woh machinery Euler-Lagrange Equations hai.

"Minus" kyun? Combination $T-V$ ($T+V$ nahi) wahi hai jiska variation sahi equations of motion deta hai — least action ke principle ka ek gehri baat. Is foundations page ke liye, bas $L=T-V$ ko recipe ki tarah maano; parent note ise ek black box ki tarah use karta hai.

7. Euler–Lagrange operator $\frac{d}{dt}\frac{\partial L}{\partial {\dot{q}}_k}-\frac{\partial L}{\partial q_k}$

Topic ko yeh kyun chahiye: ek free coordinate ke liye, is machine ko zero set karna hi motion ka niyam hai. Ek constraint ke neeche, yeh zero nahi hota — yeh bacha hua push ke barabar hota hai, constraint force $\lambda \partial f/\partial q_k$. Toh poori parent equation padhti hai: (momentum ka rate) − (applied force) = (constraint force), jo sirf $ma=$ (real forces) ko rewrite karna hai.

8. Show ka star: multiplier $\lambda$

Yeh kyun exist karna chahiye — saafs logic: §5 se, allowed nudge $\delta q$, $\nabla f$ ke perpendicular hai. d'Alembert se, wahi $\delta q$ Euler–Lagrange bracket ke perpendicular hai. Do arrows jo ek hi flat plane ke perpendicular hain parallel hone chahiye — toh bracket $=\lambda \times \nabla f$ kisi number $\lambda$ ke liye. Woh number exist karne par majboor hai; hum bas ise naam dete hain.

9. Counting: $n$, $m$, aur "balanced equations"

Yeh kyun matter karta hai: classic panic — "$\lambda$ undetermined lagta hai!" — gayab ho jaata hai jab tum count karte ho. Constraint equation $f=0$ khud woh extra relation hai jo $\lambda$ ko pin down karta hai. Bookkeeping le aao aur koi variable kabhi mat khona.

Prerequisite map

Equipment checklist

Reveal karne se pehle jawab zor se bolo. Agar koi ruk jaaye, woh section dobara padho.