Foundations — Calculus of variations — functionals, functional derivative

4.10.12 · D1 · Maths › Advanced Topics (Elite Level) › Calculus of variations — functionals, functional derivative

Is page mein kuch bhi assume nahi kiya gaya. Agar koi symbol parent note par dikh raha hai aur aap 100% sure nahi hain uska matlab kya hai, toh woh yahan define kiya gaya hai, order mein, har ek ek picture se anchor karke.

0. Function actually hoti kya hai (hamara starting stone)

Ek fact jo hum baar baar use karenge: curve ke sabse neeche wale point par tangent line flat hoti hai — uski slope zero hoti hai. Yeh "slope $=0$" poore subject ka seed hai.

• YEH TOPIC KO KYU CHAHIYE: calculus of variations poori tarah ek analogy hai — "bottom par flat slope," lekin points ki jagah curves-of-curves ke liye. Agar slope zero at a minimum ka picture solid hai, toh baaki sab translation hai.

1. Derivative $f^{\prime }(x)$ aur slope

• Picture: upar wali figure mein, valley floor par flat orange line ki slope $0$ hai; kahin aur tilted line ki non-zero slope hai.
• YEH TOPIC KO KYU CHAHIYE: "$f^{\prime }(x)=0$ at a minimum" woh sentence hai jise poora subject upgrade karta hai. Hum $\partial L/\partial y$ aur $d/dx$ symbols dekhenge — dono derivative ke types hain, isliye yeh parent page ke har symbol ka ancestor hai.

2. Derivative ke do flavours: partial $\partial$ vs total $d$

Parent page dono $\frac{\partial L}{\partial y}$ aur $\frac{d}{dx}$ likhta hai. Yeh alag tools hain. Inhe mix mat karo.

• YEH TOPIC KO KYU CHAHIYE: Euler–Lagrange equation literally partial minus total hai: $\frac{\partial L}{\partial y}-\frac{d}{dx}\frac{\partial L}{\partial y^{\prime }}$. Agar aap dono mein fark nahi kar sakte, equation gibberish hai.

3. Integral ${\int }_a^b$ — poori curve ko add karna

• Picture: neeche wali figure mein shaded band woh running total hai jo integral collect karta hai.
• YEH TOPIC KO KYU CHAHIYE: ek functional poori curve ke saath cost add karta hai — length, time, energy. Woh "curve ke saath add karna" exactly ek integral hai. Koi integral nahi, koi functional nahi.

4. Unknown ab ek curve hai: $y(x)$, $y^{\prime }(x)$, aur endpoints

• YEH TOPIC KO KYU CHAHIYE: hum un saari wires mein se choose kar rahe hain jo pins par start aur end hoti hain. Ends ko fix karna wahi hai jo baad mein ek irritating boundary term ko vanish karta hai (Section 6).

5. Lagrangian $L$ aur functional $J[y]$

• YEH TOPIC KO KYU CHAHIYE: yahi woh object hai jo hum minimise karte hain. Parent page par baaki sab kuch exist karta hai iss sawaal ka jawab dene ke liye — "kaun si curve $y$ $J[y]$ ko sabse chhota banati hai?"

6. Wiggle: $\epsilon$, test function $\eta (x)$, aur $\Phi (\epsilon )$

Yeh genuinely naya idea hai. Is picture ko master karo aur parent ki derivation inevitable ho jaati hai.

• ENDPOINTS KYU MATTER KARTE HAIN AB: integration by parts (parent, Section 2) ek boundary term $[L_{y^{\prime }}\eta {]}_a^b$ bahar nikalta hai. Kyunki $\eta (a)=\eta (b)=0$, yeh term $0$ hai aur drop ho jaati hai — yahi wajah hai ki humne fixed endpoints demand ki thi.

7. Do named results jin par parent lean karta hai

• YEH TOPIC KO KYU CHAHIYE: yeh lemma hi parent ko "$\int (\dots )\eta dx=0$ for all $\eta$" se actual Euler–Lagrange equation "$(\dots )=0$" tak jaane deta hai. Iske bina hum ek integral ke saath stuck rehte, equation nahi milti.

Prerequisite map

Yeh aage kahan feed karte hain: yahan ki machinery Lagrangian mechanics, Geodesics and differential geometry, Brachistochrone problem ko power karti hai, aur Constrained optimization & Lagrange multipliers, Ordinary differential equations (Euler–Lagrange equation ek ODE hi hai), aur Functional analysis ke broader setting se connect karti hai.

Equipment checklist

Right side cover karo; aap parent note ke liye ready hain sirf agar har reveal aapke apne words se match kare.