Foundations — Euler-Lagrange equation — derivation

4.10.13 · D1 · Maths › Advanced Topics (Elite Level) › Euler-Lagrange equation — derivation

Is page ko assume karta hai ki tumne kuch bhi nahi dekha. Hum har letter, slash, aur squiggle build karte hain jo parent Euler–Lagrange derivation use karta hai, ek aisi order mein jahan har piece apne pehle wale piece par tikti hai. Kuch bhi skip mat karo — parent fast move karta hai kyunki wo trust karta hai ki yeh sab tumhare paas already hai.

0. Woh picture jo sab kuch jodhti hai

Figure dekho. Do fixed dots — ek start point aur ek end point. Unke beech tum infinitely many curves draw kar sakte ho. Har curve ko ek score milta hai (ek single number). Calculus of variations woh machine hai jo woh ek curve dhundhti hai jiska score sabse best ho. Yeh picture dimag mein rakho: jo bhi symbols aage aate hain, woh sab iske kisi na kisi hisse ka label hain.

1. Ek function $y(x)$ — ek curve, number nahi

• Seedhe alfaazon mein: "mujhe batao tum left-to-right kahaan ho ($x$), main bataunga tum kitne upar ho ($y$)."
• Picture: upar ki figure mein koi bhi ek wiggly line.
• Topic ko kyun chahiye: hum jis unknown ko solve kar rahe hain woh ek poori curve $y(x)$ hai, ek single number nahi. Yeh school calculus se bada leap hai.

Letters $a$ aur $b$ $x$-axis par fixed endpoints mark karte hain. Jo do heights hum insist karte hain woh $y(a)=y_a$ aur $y(b)=y_b$ likhi jaati hain — "curve yahaan se start honi chahiye aur wahaan khatam honi chahiye".

2. Slope $y^{\prime }$ — curve kitni steep hai

• Seedhe alfaazon mein: curve par khado, $dx$ size ka ek infinitesimally small step right mein lo; tum $dy$ se upar jaate ho. Ratio $dy/dx$ steepness hai.
• Picture: figure mein curve se chipka hua chhota right triangle — horizontal leg $dx$, vertical leg $dy$. Iska slant tangent line hai.
• Topic ko kyun chahiye: arc length, travel time, aur energy sab depend karte hain curve kitni steep hai, isliye curve ka score $y^{\prime }$ par bhi depend karta hai jaise $y$ par. Isliye Lagrangian mein $y^{\prime }$ slot hota hai.

3. Integral ${\int }_a^b(\cdots )dx$ — running total add karna

• Seedhe alfaazon mein: "$a$ se $b$ tak har sliver mein height × width ka sum."
• Picture: figure mein curve ke neeche thin blue rectangles ka sum; jaise jaise woh patli hoti jaati hain staircase smooth shaded region ban jaata hai.
• Topic ko kyun chahiye: ek curve ka score ek running total hai jo uske saath-saath accumulate hota hai — total length, total time, total energy. Accumulation exactly wahi hai jo integral karta hai. Symbol $\int$ ek stretched "S" hai "Sum" ke liye.

Parts:

• $\int$ — summation sign.
• $a,b$ — sum kahaan se shuru hota hai aur kahaan khatam hota hai (limits).
• $dx$ — har infinitesimal strip ki width.
• $\int$ aur $dx$ ke beech ki cheez — integrand, woh height jo sum ki ja rahi hai.

4. Lagrangian $L(x,y,y^{\prime })$ — score-per-step rule

• Seedhe alfaazon mein: curve par kisi bhi point par, $L$ kehta hai "yahaan hone ki, itna upar hone ki, itna tilted hone ki local price".
• Picture: sochlo curve ki har tiny strip par ek colour painted hai — dark = mehenga, light = sasta. $L$ woh colour rule hai; integral sab colours ko add karta hai.
• Topic ko kyun chahiye: yeh woh single object hai jo tum choose karte ho apna problem encode karne ke liye. Shortest path? $L=\sqrt{1+y^{\prime 2}}$. Fastest slide? $L=\sqrt{(1+y^{\prime 2})/y}$. $L$ badlo, question badlo.

5. Do tarah ke derivatives: $\frac{\partial }{\partial y}$ vs $\frac{d}{dx}$

• Picture: partial = mixing desk par ek dial ghoomana baaki sabko tape se band karke. Total = master fader push karna; har channel jo usse wired hai woh bhi move karta hai.
• Topic ko kyun chahiye: Euler–Lagrange equation mein literally DONO hain — $\frac{\partial L}{\partial y}$ (freeze) minus $\frac{d}{dx}\frac{\partial L}{\partial y^{\prime }}$ (chain-react). Unhe confuse karna #1 error hai, isliye hum unhe yahaan alag karte hain.

Curly $\partial$ ("partial d") signal karta hai "sirf ek slot". Straight $d$ signal karta hai "chain rule chalne do".

6. Functional $J[y]$ — ek machine jo curve khaata hai, number ugalta hai

• Seedhe alfaazon mein: "mujhe ek curve do, main uska score integrate karke total tumhein dunga."
• Picture: upar ki poori figure — har candidate curve box $J[\cdot ]$ mein jaati hai aur ek scoreboard number bahar aata hai.
• Topic ko kyun chahiye: $J[y]$ wahi quantity hai jo hum minimize karte hain. Ordinary calculus $f(x)$ ko numbers par minimize karta hai; variations $J[y]$ ko curves par minimize karta hai. Square brackets us jump ka visual reminder hain.

7. Variation: $\eta (x)$, $\epsilon$, aur perturbed curve $Y$

• Seedhe alfaazon mein: true curve $y$ lo, $\epsilon$ times koi bump $\eta$ add karo. $\epsilon \to 0$ karo aur bump gayab ho jaata hai — tum wapas true curve par ho.
• Picture: figure mein amber true curve plus dashed nearby curves, har ek alag $\epsilon$ ke liye $y+\epsilon \eta$.
• Topic ko kyun chahiye: "kya yeh curve optimal hai?" test karne ke liye, tumhe poori curve nudge karne aur score check karne ki zaroorat hai. $\eta$ aur $\epsilon$ wahi nudge hain.

Kyunki har competitor abhi bhi fixed endpoints hit karna chahiye, wiggle ends par khatam ho jaani chahiye: $\eta (a)=\eta (b)=0.$ Figure mein dashed curves dono dots par wapas pinch ho jaati hain — woh pinching hi yeh condition hai. Yeh exactly wahi hai jo baad mein derivation mein boundary term ko khatam karta hai.

8. Stationary — "first order par flat"

• Seedhe alfaazon mein: tum wahaan khade ho jahan ground momentarily level hai; kisi bhi direction mein ek baal bhar step lo aur, first approximation mein, tum na upar jaate ho na neeche.
• Picture: ek valley ka flat bottom, ya ek hill ka flat top.
• Topic ko kyun chahiye: hum true minimum demand NAHI karte — sirf itna ki $J$ stationary ho. Yahi "$=0$" hai ${\frac{d\Phi }{d\epsilon }|}_{\epsilon =0}=0$ mein, jahan $\Phi (\epsilon )=J[y+\epsilon \eta ]$ woh ordinary function hai jo score ban jaata hai jab hum ek wiggle fix karte hain.

9. Integration by parts — factors ke beech derivative trade karna

• Seedhe alfaazon mein: agar ek factor derivative carry karta hai, tum woh derivative doosre factor par move kar sakte ho, boundary term $[uv{]}_a^b$ ki cost par.
• Topic ko kyun chahiye: derivation mein wiggle ${\eta }^{\prime }$ ke roop mein appear hoti hai (differentiated). Hume ek bare $\eta$ chahiye taaki hum use factor out kar sakein. Integration by parts derivative ko $\eta$ se hatata hai aur $\frac{\partial L}{\partial y^{\prime }}$ par dalta hai, aur bacha hua boundary term mar jaata hai kyunki $\eta (a)=\eta (b)=0$. Full drill ke liye Integration by Parts dekho.

Prerequisite map

Ise upar se neeche padho: raw ideas (function, slope, integral) Lagrangian aur functional ko feed karte hain; wiggle plus stationarity plus integration by parts Euler–Lagrange equation mein assemble hote hain. Sab kuch parent derivation mein route hota hai, jo phir Lagrangian Mechanics, Brachistochrone Problem, Geodesics, Beltrami Identity, aur Noether's Theorem ko power karta hai — sab Principle of Least Action aur Calculus of Variations par built hain.

Equipment checklist

Cover the right side and answer aloud before revealing.