Foundations — Hamilton's principle — least action

4.10.15 · D1 · Maths › Advanced Topics (Elite Level) › Hamilton's principle — least action

Parent note Hamilton's Principle bohot saare symbols ek saath throw karta hai: $q$, $\dot{q}$, $L$, $T$, $V$, $\int$, $\delta$, $\partial$, $\eta$, $\epsilon$. Yeh page har ek ko absolute zero se build karta hai, uss order mein jisme ek doosre ki zaroorat padti hai. Agar aap line one padh sakte ho, toh sab padh sakte ho.

1. Position, lekin flexible: coordinate $q$

Picture. Ek bent wire par ek bead socho. Tumhe $x$ aur $y$ dono ki zaroorat nahi — ek number "wire ke along kitni door" already sab kuch bata deta hai. Woh ek number $q$ hai.

Topic ko iske ki zaroorat kyun. Forces $x,y,z$ mein rehti hain aur tumhe unhe components mein chop karna padta hai. Ek accha chosen $q$ (jaise pendulum ka angle $\theta$) geometry ko absorb kar leta hai toh tumhe kuch bhi decompose nahi karna padta. Yahi poore subject ka practical payoff hai.

2. Coordinate ki speed: $\dot{q}$ aur dot

Picture. Agar $q$ ek runner ki track par position hai, toh $\dot{q}$ speedometer reading hai, aur $\ddot{q}$ batata hai ki woh accelerate kar raha hai ya brake maar raha hai.

$\dot{q}=\frac{dq}{dt},\ddot{q}=\frac{d^{2}q}{d{t}^2}.$

Notation kyun. Physicists dot specifically time derivatives ke liye likhte hain, prime ${}^{\prime }$ ko space derivatives ke liye free karte hain. $L(q,\dot{q},t)$ mein system ki "state" ke liye dono chahiye — kahan hai ($q$) aur kitni tezi se move kar raha hai ($\dot{q}$) — sirf position se kinetic energy nahi bata sakte.

3. Do tarah ki change: $T$ aur $V$ (energy)

Picture. Ek ball jo valley mein roll kar rahi hai. Valley ke andar deep mein fast move karti hai ⇒ bada $T$, chhota $V$. Slope ke top par ruk jaati hai ⇒ bada $V$, chhota $T$. Jaise roll karti hai, energy dono buckets ke beech slosh karti rehti hai.

Topic ko dono ki zaroorat kyun. Poori machine difference $T-V$ se bani hai. Subtract karne se pehle humhe dono buckets alag chahiye.

4. Show ka star: Lagrangian $L$

Picture. Clock ki har tick par ball ka kuch $T$ aur kuch $V$ hota hai; subtract karo ek number $L$ milta hai. Jaise ball move karti hai, $L$ time ke saath ek curve trace karta hai.

$L$ ko $L(q,\dot{q},t)$ kyun likha jaata hai. Depend karta hai kahan ho ($q$, $V$ ke through), kitni tezi se move kar rahe ho ($\dot{q}$, $T$ ke through), aur possibly clock par ($t$, agar setup time mein change hoti hai). Teen slots — yaad rakho, yeh step 8 mein matter karte hain.

5. Journey mein add karna: integral $\int$

Picture. Journey ko lakho tiny time-steps mein kaato. Har step mein $L$ ki koi value hoti hai; tiny duration $dt$ (ek thin strip) se multiply karo. Saari strips stack karo ⇒ $L$-versus-$t$ curve ke neeche total shaded area.

Topic ko iske ki zaroorat kyun. Ek path ko ek instant par judge nahi kiya jaata — uski poori duration par judge kiya jaata hai. Integration precisely woh tool hai jo "har instant par value" ko "ek total score" mein collapse karta hai.

6. Score khud: action $S$ aur word functional

Picture. Ek path feed karo — start se end tak koi bhi wiggly curve — aur ek number nikalta hai. Alag path se alag number milta hai. Square brackets $S[q]$ note karo: yeh warn karte hain ki input poora function hai, sirf ek value nahi.

"Functional" kyun, "function" kyun nahi? Ek plain function ek number khaata hai aur ek number deta hai ($f(3)=9$). Ek functional ek function khaata hai. Yeh ek level upar hai, aur isi liye humhe Calculus of Variations chahiye — ordinary calculus ek number ke respect mein differentiate karta hai; yahan humhe "ek poore curve ke respect mein differentiate" karna padta hai.

7. Tiny nudge: $\eta$, $\epsilon$, aur wiggle

Best path dhundhne ke liye hum ise nearby paths se compare karte hain. Hum ek nearby path banate hain ek small bump jodke:

$q(t)⟶q(t)+\epsilon \eta (t).$

Picture. True path lo aur beech mein dhyaan se guitar string ki tarah pluck karo, dono ends fixed rakho. Pluck ki shape $\eta$ hai; kitni tezi se pluck kiya woh $\epsilon$ hai. Kyunki ends nail down hain, har trial path abhi bhi same do points par start aur finish karti hai.

Ends par pin kyun. Hamilton's principle poochta hai "un saari paths mein se jinka start aur finish same hai, kaun si best hai?" Hum endpoints move nahi kar sakte, isliye nudge wahan vanish karna chahiye. (Yahi pinning ek leftover term baad mein kill karti hai — Euler–Lagrange equation dekho.)

8. Derivative ke do flavours: $\partial$ vs $\frac{d}{dt}$

Picture. $L$ teen dials wale dashboard jaisa hai: $q$, $\dot{q}$, $t$. Partial derivative ek dial ghoomata hai aur $L$ dekhta hai. Total time-derivative clock chalata hai aur saare dials ek saath move karte hain (kyunki $q$ aur $\dot{q}$ dono time ke saath drift karte hain).

Topic ko dono ki zaroorat kyun. Euler–Lagrange equation $\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}}\right)-\frac{\partial L}{\partial q}=0$ dono ko mix karti hai: pehle partial lo ($\dot{q}$-slot isolate karo), phir total time derivative lo (use evolve hone do). Dono ko confuse karna sabse common galti hai.

9. Stationary condition: $\delta S=0$

Picture. Action-score ko ek landscape ki height socho, jahan ground ka har point ek poora path hai. True path ek flat point par hai — valley bottom, hilltop, ya saddle — jahan ground har direction mein level hai. Wahan rakhi ball nahi roll karegi.

Foundations topic ko kaise feed karte hain

Har foundation Euler–Lagrange equation ka prerequisite hai, jo baad mein parent Hamilton's Principle ko power karta hai. Wahan se tum Lagrangian Mechanics, Noether's Theorem, Hamiltonian Mechanics, Fermat's Principle mein branch kar sakte ho, aur dekh sakte ho yeh sab kaise Newton's Laws par reduce hota hai.

Equipment checklist

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