Foundations — Applying E-L equations to various systems

2.1.6 · D1 · Physics › Analytical Mechanics › Applying E-L equations to various systems

Isse pehle ki tum parent recipe use kar sako, iske har squiggle ka matlab kuch concrete hona chahiye. Yeh page har ek ko zero se build karta hai, us order mein jisme woh ek doosre pe depend karte hain. Agar neeche koi term pehle se obvious lagti hai, toh skip karo — lekin yahan kuch bhi assumed nahi hai.

1. Coordinate kya hota hai?

Picture: ek bead ek seedhi wire pe threaded hai. Bead kahan hai yeh batane ke liye tumhe exactly ek number chahiye — wire pe woh kitni door baitha hai. Woh number ek coordinate hai.

Topic ko yeh kyun chahiye: poora method "coordinates chuno" se shuru hota hai. Agar tum nahi jaante coordinate kya hota hai, toh recipe ka step 1 ek black box hai.

Notice karo ki do alag bead-on-wire pictures same idea deti hain: ek number = ek position. Number ek distance ho sakta hai (rod pe metres) ya ek angle (circle ke around degrees). Dono coordinates hain.

2. Degrees of freedom (DOF)

Picture:

• Ek straight wire pe bead → 1 DOF (ek number: kitni door).
• Flat table pe ek free dot → 2 DOF (do number: left-right aur up-down).
• Room mein float karta ek free dot → 3 DOF (teen number).

Topic ko yeh kyun chahiye: parent ka step 1 hai "DOF gino → utne hi coordinates chuno." Kam coordinates se motion describe nahi hogi; zyada se woh independent nahi rahenge. DOF tumhe exact sahi count batata hai.

3. Generalized coordinates $q_i$

Notation padhna: letter $q$ "kisi coordinate" ke liye stand-in hai. Chhota subscript $i$ ek label hai — $q_1,q_2,q_3,\dots$ — ek $q$ har degree of freedom ke liye. Agar 2 DOF hain toh tumhe $q_1$ aur $q_2$ dikhenge.

Picture: wahi pendulum bob, do tarike se describe kiya.

Topic ko yeh kyun chahiye: sabse clever coordinate choose karne ki freedom is method ka pura payoff hai. Pendulum ke liye $\theta$ choose karna ya incline ke saath $s$ — yahi hai jo constraint force ko vanish karta hai. Symbol $q_i$ deliberately generic hai taaki ek recipe yeh sab choices ek saath cover kare.

4. Dot: $\dot{q}$ ka matlab "rate of change" hai

Dot kyun aur kuch aur kyun nahi? Yeh sirf $\frac{dq}{dt}$ ka shorthand hai — "$q$ mein change, time mein change se divided". Physicists dot likhte hain kyunki time-derivatives har line pe aate hain aur $\dot{q}$, $\frac{dq}{dt}$ se zyada fast hai.

Picture: agar $q=$ pendulum ka angle hai, toh $\dot{q}$ yeh hai ki angle kitni tezi se sweep hota hai — swing ki angular speed. Do dots, $\ddot{q}$, matlab rate of change of the rate of change — acceleration.

Topic ko yeh kyun chahiye: Lagrangian position $q$ aur uski speed $\dot{q}$ dono pe depend karta hai, aur motion ki final equation acceleration $\ddot{q}$ ke baare mein hai. Dot ke bina tum $L(q,\dot{q},t)$ padh bhi nahi sakte.

5. Kinetic energy $T$

Letter $T$ kyun? Historical convention (purani mechanics texts se). Iska time ya temperature se koi lena-dena nahi — iska matlab sirf kinetic energy hai. Iske aadat daalo, kyunki parent har jagah $T$ use karta hai.

Picture: ek bhaari ball ya ek tez ball ko rokna mushkil hai — zyada $T$. Kyunki $v$ squared hai, speed double karne se energy chaar guna ho jaati hai.

Topic ko yeh kyun chahiye: $T$ energy score ka aadha hissa hai. Har example pehle $T$ compute karta hai.

6. Potential energy $V$

Picture: ek kitaab ko shelf pe uthao — tumne usme energy store kar di; chodo aur gravity us store ko wapas motion (kinetic energy) mein convert kar deti hai. Zyada upar = zyada stored $V$.

Zero tumhari choice hai. Sirf $V$ mein differences matter karte hain, isliye tum decide karo ki "$V=0$" kahan hai. Pendulum mein hum lowest point ko zero set karte hain, jo deta hai $V=mg\ell (1-\cos \theta )$ (zero neeche, maximum upar).

Topic ko yeh kyun chahiye: $V$ energy score ka doosra aadha hissa hai. Poora Lagrangian hai $L=T-V$.

7. Lagrangian $L=T-V$

Minus kyun, plus kyun nahi? Total energy hai $T+V$. Lagrangian ka minus sign accident nahi hai — yeh exactly woh combination hai jiska "least action" Newton's laws reproduce karta hai. Jab tum 2.1.05-Deriving-the-Euler-Lagrange-equation padhoge tab deep reason samajh aayega; abhi ke liye $L=T-V$ ko definition ke roop mein accept karo.

8. Partial derivatives $\frac{\partial L}{\partial q}$ aur $\frac{\partial L}{\partial \dot{q}}$

Picture: $L$ ko ek landscape ki height samjho, jisme do ground directions $q$ (east) aur $\dot{q}$ (north) label hain. East mein chalna eastward slope $\partial L/\partial q$ measure karta hai; north mein chalna $\partial L/\partial \dot{q}$ measure karta hai. Har partial ek chosen direction mein steepness hai.

Topic ko yeh kyun chahiye: E–L equation poori tarah inhi do partials se bani hai. Agar tum inhe compute nahi kar sakte, toh recipe apply nahi ho sakti.

9. Full time-derivative $\frac{d}{dt}$

Partial se fark: partial doosre variables freeze karta hai; total $d/dt$ sab ko time ke saath ride karne deta hai. E–L equation mein term $\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}}\right)$ pehle freeze-and-differentiate karta hai ($\partial$), phir un-freeze karke time flow karne deta hai ($d/dt$).

Picture: pendulum ke liye $\frac{\partial L}{\partial \dot{\theta }}=m{\ell }^2\dot{\theta }$. Isme $\frac{d}{dt}$ apply karne pe milta hai $m{\ell }^2\ddot{\theta }$ — $\dot{\theta }$ ban gaya $\ddot{\theta }$ kyunki time bahat raha tha. Yahan acceleration finally appear hoti hai.

10. Sab ek saath: Euler–Lagrange equation

Har piece upar ek section se trace hoti hai: $L$ (§7), ${\dot{q}}_i$ aur dot (§4), partials (§8), total derivative (§9), aur $q_i$ (§3). Ab isme kuch bhi mysterious nahi hai.

11. Cyclic coordinate aur conserved momentum (preview)

Picture: ek central-force problem mein $L$ mein $\dot{\theta }$ hai lekin $\theta$ khud nahi — physics ko parwah nahi ki "north" kis taraf hai. Woh symmetry tumhe ek conserved quantity (angular momentum) free mein de deti hai. Yeh symmetry aur conservation ke beech deep link Noethers-theorem hai, expand hua 2.1.07-Cyclic-coordinates-and-conservation-laws mein.

Topic ko yeh kyun chahiye: parent mein Example 4 exactly isi ko use karta hai $L_z=m{r}^2\dot{\theta }=\text{const}$ paane ke liye bina kisi extra kaam ke.

Prerequisite map

Ise upar se neeche padho: positions aur rates do energies ko feed karte hain, energies $L$ banati hain, $L$ partials ko feed karta hai, aur partials plus total time-derivative Euler–Lagrange equation assemble karte hain — poore parent topic ka engine.

Equipment checklist

Cover the right-hand side aur khud ko test karo. Agar koi answer fuzzy hai, toh parent ke examples tackle karne se pehle woh section dobara padho.

Ready ho? Wapas jao parent topic pe aur pendulum pe five-step recipe chalao.