Kinetic energy in generalized coordinates — Hinglish notes

2.1.3 · Physics › Analytical Mechanics

KYA compute kar rahe hain

Total kinetic energy hamesha yahi hai: $T = \sum_{a} \frac{1}{2} m_{a} \dot{r}_{a} \cdot \dot{r}_{a} .$ Hamara kaam: $\dot{r}_{a}$ ko $\overset{q}{˙}_{i}$ mein convert karna.

KAISE: first principles se derive karo

Step 1 — Position map differentiate karo (chain rule). Kyunki $r_{a} = r_{a} (q, t)$ hai, total time derivative hai: $\dot{r}_{a} = \sum_{i = 1}^{n} \frac{\partial r _{a}}{\partial q _{i}} \overset{q}{˙}_{i} + \frac{\partial r _{a}}{\partial t} .$

Ye step kyun? Chain rule kehta hai ki har velocity component "position per coordinate kaise change hoti hai" $\times$ "woh coordinate kitni tez change hota hai" ka sum hai, plus explicit clock term $\partial_{t} r_{a}$ jo moving constraints se aata hai.

Step 2 — $T$ mein substitute karo. $T = \sum_{a} \frac{1}{2} m_{a} (\sum_{i} \frac{\partial r _{a}}{\partial q _{i}} \overset{q}{˙}_{i} + \frac{\partial r _{a}}{\partial t}) \cdot (\sum_{j} \frac{\partial r _{a}}{\partial q _{j}} \overset{q}{˙}_{j} + \frac{\partial r _{a}}{\partial t}) .$

Ye step kyun? Hum literally dot product ko same expression mein plug karke expand karte hain. Ek sum ka square teen tarah ke terms deta hai: $\overset{q}{˙} \overset{q}{˙}$ , $\overset{q}{˙} \times (\partial_{t})$ , aur $(\partial_{t}) \times (\partial_{t})$ .

Step 3 — Velocity ke powers ke hisaab se collect karo. Product expand karne par: $T = T_{2} + T_{1} + T_{0}$ jahan

$T_{2} = \frac{1}{2} \sum_{i, j} M_{ij} (q, t) \overset{q}{˙}_{i} \overset{q}{˙}_{j}, M_{ij} = \sum_{a} m_{a} \frac{\partial r _{a}}{\partial q _{i}} \cdot \frac{\partial r _{a}}{\partial q _{j}}$

$T_{1} = \sum_{i} a_{i} (q, t) \overset{q}{˙}_{i}, a_{i} = \sum_{a} m_{a} \frac{\partial r _{a}}{\partial q _{i}} \cdot \frac{\partial r _{a}}{\partial t}$

$T_{0} = \frac{1}{2} \sum_{a} m_{a} (\frac{\partial r _{a}}{\partial t})^{2} .$

Ye step kyun? Kitne $\overset{q}{˙}$ factors hain uske hisaab se group karna sabse natural classification hai. $T_{2}$ velocities mein quadratic hai, $T_{1}$ linear hai, $T_{0}$ mein koi velocity nahi.

Crucial special case: scleronomic systems

Ek homogeneous quadratic function Euler's theorem satisfy karta hai: $\sum_{i} \overset{q}{˙}_{i} \partial T / \partial \overset{q}{˙}_{i} = 2 T$ . Isliye $\sum_{i} p_{i} \overset{q}{˙}_{i} - L$ energy deta hai aur isliye $H = T + V$ jab system scleronomic hota hai — yeh apni pocket mein rakh lo.

Worked examples

Common mistakes

Recall Feynman: ek 12-saal ke bachche ko explain karo

Socho tumhara toy car sirf bent tracks par chal sakta hai. Yeh kehne ke liye ki woh sach mein kitna tez chal raha hai, tum normally plain left-right-up-down speed use karte. Lekin yeh kehna aasaan hai ki "woh track ke saath kitna tez ja raha hai." Math hume ek ko doosre se trade karne deta hai. Agar track khud kheencha ja raha hai (koi poori table spin kar raha hai), toh track par bilkul still baitha car bhi room mein actually move kar raha hai — toh hum extra speed terms add karte hain. Yeh extra stuff hi $T_{1}$ aur $T_{0}$ pieces hain; jab koi table nahi kheenchta, woh zero hote hain aur energy simple $\frac{1}{2} (stuff) \times (speed)^{2}$ hoti hai.

Active recall

\dot{r}_{a}

mein

\partial r_{a} / \partial t

term kyun hota hai?

Kyunki rheonomic (moving) constraints ke under position map

r_{a} (q, t)

explicitly time par depend karta hai, toh chain rule ek explicit clock term add karta hai.

T

ka general decomposition likho.

T = T_{2} + T_{1} + T_{0}

—

\overset{q}{˙}_{i}

mein quadratic, linear, aur velocity-free.

Mass matrix

M_{ij}

ki definition?

M_{ij} = \sum_{a} m_{a} \partial_{q_{i}} r_{a} \cdot \partial_{q_{j}} r_{a}

; symmetric aur positive-definite.

T

pure homogeneous quadratic (

T = T_{2}

) kab hota hai?

Jab constraints scleronomic hon (

\partial r_{a} / \partial t = 0

), jisse

a_{i} = 0

aur

T_{0} = 0

ho jaata hai.

Plane polar coordinates mein

T

kya hai?

T = \frac{1}{2} m (\overset{r}{˙}^{2} + r^{2} \dot{θ}^{2})

Euler's theorem ke hisaab se

\sum_{i} \overset{q}{˙}_{i} \partial T_{2} / \partial \overset{q}{˙}_{i}

kya hai?

2 T_{2}

(velocities mein degree 2 ka homogeneous).

Rotating wire ke liye

T_{0}

ka physical meaning kya hai?

\frac{1}{2} m r^{2} ω^{2}

— ek effective potential ki tarah act karta hai jo centrifugal effect deta hai.

Kya

M_{ij}

generally constant hai?

Nahi, yeh generalized coordinates

q

par depend karta hai (e.g.

M_{θ θ} = m r^{2}

Connections

Lagrangian mechanics — $L = T - V$ is $T$ ka use karta hai.
Generalized coordinates and constraints — scleronomic vs rheonomic.
Euler-Lagrange equations — jahan $\partial T / \partial \overset{q}{˙}$ aur $\partial T / \partial q$ feed in hote hain.
Generalized momentum and conserved quantities — $p_{i} = \partial T / \partial \overset{q}{˙}_{i} = \sum_{j} M_{ij} \overset{q}{˙}_{j}$ .
Configuration space and the metric tensor — $M_{ij}$ ek Riemannian metric ke roop mein.
Energy function and Hamiltonian — generally $H = T_{2} - T_{0} + V$ ; $H = T + V$ sirf tab jab scleronomic ho.