Calculus of variations — functionals, functional derivative

4.10.12 · Maths › Advanced Topics (Elite Level)

1. Functional KYA hota hai?

YEH KYUN MATTER KARTA HAI: Physics aur geometry mein bahut saare problems "sabse achha curve dhundo" wale hote hain:

• Do points ke beech sabse chhota raasta (geodesic): $L=\sqrt{1+y^{\prime 2}}$.
• Brachistochrone (sabse tez slide): $L=\sqrt{\frac{1+y^{\prime 2}}{2gy}}$.
• Classical mechanics: $L=T-V$, action $S=\int Ldt$ minimise hota hai (Hamilton's principle).

2. Functional derivative — scratch se derive karte hain

CHAHIYE KYA: "$\frac{df}{dx}=0$" ka functionals ke liye analogue.

KAISE — variation trick. Maano $y(x)$ minimiser hai, fixed endpoints $y(a),y(b)$ ke saath. Ise thoda perturb karo: $\tilde{y}(x)=y(x)+\epsilon \eta (x),\eta (a)=\eta (b)=0.$ Yahan $\eta$ koi bhi smooth "test function" hai jo ends par vanish karti hai (taaki endpoints fixed rahe). Define karo: $\Phi (\epsilon )=J[y+\epsilon \eta ].$ Kyunki $y$, $J$ ko minimise karta hai, isliye ordinary function $\Phi$ ka minimum $\epsilon =0$ par hai, toh: $\boxed{{\Phi }^{\prime }(0)=0}\text{(yahi hai neeche sab kuch ka WHY).}$

${\Phi }^{\prime }(0)$ compute karo. Integral ke andar differentiate karo: ${\Phi }^{\prime }(\epsilon )={\int }_a^b(\frac{\partial L}{\partial y}\eta +\frac{\partial L}{\partial y^{\prime }}{\eta }^{\prime })dx.$ Yeh step kyun? Humne $L(x,y+\epsilon \eta ,y^{\prime }+\epsilon {\eta }^{\prime })$ par chain rule use kiya; $\partial /\partial \epsilon$ $y$-slot ke liye $\eta$ aur $y^{\prime }$-slot ke liye ${\eta }^{\prime }$ le aata hai.

$\epsilon =0$ set karo aur doosre term ko integration by parts se $\eta$ ko uske derivative se free karo: ${\int }_a^b\frac{\partial L}{\partial y^{\prime }}{\eta }^{\prime }dx=\underset{=0(\eta (a)=\eta (b)=0)}{\underbrace{[\frac{\partial L}{\partial y^{\prime }}\eta {]}_a^b}}-{\int }_a^b\frac{d}{dx}\left(\frac{\partial L}{\partial y^{\prime }}\right)\eta dx.$ Yeh step kyun? Integration by parts derivative ko $\eta$ se hatake $L_{y^{\prime }}$ par le jaata hai. Boundary term khatam ho jaata hai kyunki $\eta$ endpoints par vanish karta hai — isi liye humne fixed endpoints ki requirement rakhi thi.

Toh: ${\Phi }^{\prime }(0)={\int }_a^b\underset{\text{ise kehte hain }\frac{\delta J}{\delta y}}{\underbrace{[\frac{\partial L}{\partial y}-\frac{d}{dx}\frac{\partial L}{\partial y^{\prime }}]}}\eta (x)dx=0\forall \eta .$

Fundamental Lemma. Agar ${\int }_a^{b}f(x)\eta (x)dx=0$ har smooth $\eta$ ke liye jo ends par vanish kare, toh $f(x)\equiv 0$. Kyun sach hai: agar $f>0$ kahin hai, toh wahan concentrated ek bump $\eta$ choose karo jo integral ko positive bana de — contradiction.

3. Ek useful shortcut: Beltrami identity

Agar $L=L(y,y^{\prime })$ explicitly $x$ par depend na kare, toh ek first integral exist karta hai: $\frac{d}{dx}(L-y^{\prime }\frac{\partial L}{\partial y^{\prime }})=L_y{y}^{\prime }+L_{y^{\prime }}{y}^{\prime \prime }-y^{\prime \prime }{L}_{y^{\prime }}-y^{\prime }\frac{d}{dx}{L}_{y^{\prime }}=y^{\prime }(L_y-\frac{d}{dx}{L}_{y^{\prime }})=0.$ Yeh step kyun? Total $x$-derivative expand karo; $y^{\prime \prime }$ terms cancel ho jaate hain, aur bracket exactly Euler–Lagrange expression hai $=0$. Isliye:

4. Worked examples

5. Common mistakes

6. Active recall

Connections

• Lagrangian mechanics — action $S=\int Ldt$, E–L Newton's laws deta hai.
• Geodesics and differential geometry — variational problems se shortest paths.
• Functional analysis — Gâteaux/Fréchet derivatives $\delta J/\delta y$ ko generalise karte hain.
• Constrained optimization & Lagrange multipliers — isoperimetric problems functionals par multipliers use karte hain.
• Ordinary differential equations — E–L wahi ODE hai jo tumhe actually solve karni hoti hai.
• Brachistochrone problem — historic origin (Bernoulli, 1696).

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