Kinetic energy in generalized coordinates: Foundations

2.1.3 · D1 · Physics › Analytical Mechanics › Generalized coordinates mein kinetic energy

Yeh ground floor hai. Agar parent note mein kabhi koi aisa symbol likha ho jo tum zor se bol na sako, toh woh yahan hai — defined aur drawn.

1. Ek particle, uski position, aur arrow $r$

Picture: kamre ke ek chosen kone se ball tak ek arrow. Figure s01 dekho — arrow $r$ aur uske teen shadow-lengths $x, y, z$ deewaaron par.

Figure — Kinetic energy in generalized coordinates

Yeh topic ko kyun chahiye: aage ka saara kaam yahi hai — "particle kahan hai, aur woh arrow kitni tezi se change ho raha hai?" Jab tak hum position point nahi kar sakte, energy ki baat nahi kar sakte.

Subscript $r_{a}$ mein bas yeh batata hai ki kaun sa particle hai: $r_{1}$ particle 1 hai, $r_{2}$ particle 2, aur aise hi $r_{N}$ tak $N$ particles ke liye. Bold typeface ka matlab hamesha yeh hai ki "yeh ek arrow (vector) hai, single number nahi".

2. Dot $˙$ — velocity ek rate of change ke roop mein

Dot kyun, $d / d t$ kyun nahi? Purely shorthand hai — mechanics mein almost har derivative time ke saath hoti hai, isliye Newton ka dot ink bachata hai. $\ddot{r}$ (do dots) acceleration hoga, lekin parent topic ko uski zaroorat nahi.

Picture: particle ko ek instant par roko, phir ek heartbeat baad; dono positions ko join karne wala tiny arrow, us tiny time se divide karna, woh $\dot{r}$ hai. Yeh direction of travel ki taraf point karta hai aur iska length speed hai.

3. Dot product $\cdot$ — do arrows ko ek number mein banana

Special case jo hum sabse zyada use karte hain woh hai ek arrow ko khud se dot karna: $v \cdot v = v_{1}^{2} + v_{2}^{2} + v_{3}^{2} = ∣ v ∣^{2} = v^{2},$ length-squared, yani speed-squared. Isliye kinetic energy $T = \frac{1}{2} m \dot{r} \cdot \dot{r}$ likhi jaati hai — yeh $\frac{1}{2} m v^{2}$ hai lekin vector ke kapdon mein.

Picture: figure s02 do arrows dikhata hai aur woh do cheezein jo dot product measure karta hai — ek dusre ke along kitna hai, aur (jab equal ho) plain squared length.

4. Kinetic energy $T$ aur $\frac{1}{2} m v^{2}$ jahan se hum shuru karte hain

Picture: har ball ek chota energy bag carry karti hai jo uski mass times speed-squared ke proportional hai; $T$ matlab saare bags ek dher mein khaali karna. Yahan kuch bhi mysterious nahi — yeh Newtonian energy hai. Parent note ki poori kaala $\dot{r}_{a}$ ko better coordinates mein rewrite karna hai.

5. Generalized coordinates $q_{i}$ — "natural" description

$n$ kyun, $3 N$ kyun nahi? Ek single free particle ko 3 numbers chahiye; $N$ particles ko $3 N$ . Lekin constraints (ek bead wire par phansa hua, fixed length ka pendulum) unme se bade hisson ko forbid karte hain. $n$ — jo actually survive karta hai — aksar bahut chhota hota hai. Dekho Generalized coordinates and constraints.

Picture: figure s03 — ek bent wire par ek bead. Uski poori room-position ke liye $x, y$ chahiye, lekin wire ke saath ek number $q$ (arc length) sab kuch bata deta hai. Woh ek number ek generalized coordinate hai.

Index $i$ in $q_{i}$ bas yeh label karta hai ki hum kaun si generalized coordinate ki baat kar rahe hain; $\overset{q}{˙}_{i}$ (dot ke saath) us coordinate ka rate of change hai — "us natural direction ke saath speed".

6. Position map $r_{a} (q, t)$ — worlds ke beech dictionary

Extra slot $t$ (time) tabhi appear karta hai jab ek constraint khud baar bahar se impose kiye schedule par move ho raha hoo — ek wire jise koi spin kar raha hai, ek support jo raise ho raha hai. Aisi constraint ko rheonomic (moving) kehte hain. Agar koi cheez constraint ko bahar se move nahi kar rahi, toh $t$ absent hai aur constraint scleronomic (rigid, time-frozen) hai.

Picture: do dials ( $q$ aur, agar present ho, ek clock $t$ ) jo ek machine mein feed ho rahe hain jo room-arrow $r_{a}$ bahar spits karta hai.

7. Partial derivatives $\frac{\partial r _{a}}{\partial q _{i}}$ aur $\frac{\partial r _{a}}{\partial t}$

Seedha- $d$ derivative $d / d t$ sab kuch ek saath change karta hai; curly- $\partial$ derivative ek cheez change karta hai, baaki sab fixed rakhke. Hume curly wala chahiye kyunki hamare map mein kai inputs hain.

Do flavours matter karte hain:

$\frac{\partial r _{a}}{\partial q _{i}}$ — motion jo tumhare coordinate $i$ ghuma'ne se hoti hai.
$\frac{\partial r _{a}}{\partial t}$ — motion jo constraint ke khud move karne se hoti hai (sirf rheonomic systems ke liye nonzero).

Topic ko dono kyun chahiye: particle ki velocity us "motion ka sum hai jo main $q_{i}$ ke through command karta hoon" plus "motion jo moving constraint mujh par force karti hai". Woh split literally parent note ka Step 1 hai.

8. Chain rule — velocity kaise assemble hoti hai

Ise ek sentence ki tarah padho: total room-velocity $=$ (har dial ke liye: "yeh mujhe kis taraf push karta hai" $\times$ "main use kitni tezi se ghuma raha hoon") sum up, $+$ ("moving constraint mujhe kis taraf drag karti hai").

9. Double sums aur mass matrix $M_{ij}$

Jab hum chain-rule velocity ko square karte hain, toh har pair $(i, j)$ ke liye products $\overset{q}{˙}_{i} \overset{q}{˙}_{j}$ appear karte hain. $\sum_{i, j}$ likhna matlab hai "saare ordered pairs $i$ aur $j$ par sum karo". Har aisi term ek coefficient carry karti hai: $M_{ij} = \sum_{a} m_{a} \frac{\partial r _{a}}{\partial q _{i}} \cdot \frac{\partial r _{a}}{\partial q _{j}},$ numbers ki ek table (matrix) jo do labels se indexed hai.

$M_{ij}$ symmetric hai ( $M_{ij} = M_{j i}$ , kyunki dot product order ki parwah nahi karta) aur positive-definite (real motion hamesha positive energy cost karti hai).

Prerequisite map

Har arrow ka matlab hai "tail samajhne ke baad hi head make sense karega." Do streams — energy stream (left) aur coordinate stream (right) — topic par hi milte hain.

Equipment checklist

Apne aap ko test karo: right side cover karo, answer do, reveal karo.

Arrow

r

physically kya represent karta hai?

Fixed origin se particle tak ka arrow; uske teen numbers right/forward/up distances hain.

Overdot ka matlab kya hai, jaise

\dot{r}

mein?

Rate of change per second — yahan velocity, yani position arrow kitni tezi se move kar raha hai.

v = (3, 4, 0)

ke liye

v \cdot v

compute karo.

9 + 16 + 0 = 25

, jo speed-squared

v^{2}

hai (toh speed

= 5

Kinetic energy likhne ke liye hum dot product kyun use karte hain?

Kyunki

\dot{r} \cdot \dot{r} = v^{2}

direction-blind speed-squared deta hai — exactly woh jo

\frac{1}{2} m v^{2}

ko chahiye.

Ek line mein, generalized coordinate kya hai?

Koi bhi independent number (length YA angle YA arc-length) jo system ka arrangement fix karne mein help karta hai, constraints already account hoke.

Position map

r_{a} (q, t)

mein

t

explicitly kab hota hai?

Sirf rheonomic constraints ke liye — jab koi cheez constraint ko bahar se schedule par move karti hai.

d / d t

aur

\partial / \partial q_{i}

mein kya fark hai?

d / d t

sab kuch ek saath change karta hai;

\partial / \partial q_{i}

sirf

q_{i}

ko nudge karta hai aur baaki sab inputs freeze rakhta hai.

\dot{r}_{a}

ke liye chain rule state karo.

\dot{r}_{a} = \sum_{i} (\partial r_{a} / \partial q_{i}) \overset{q}{˙}_{i} + \partial r_{a} / \partial t

Mass matrix

M_{ij}

ki do properties kya hain?

Symmetric (

M_{ij} = M_{j i}

) aur positive-definite.

T_{2} = \frac{1}{2} \sum_{i, j} M_{ij} \overset{q}{˙}_{i} \overset{q}{˙}_{j}

mein

\frac{1}{2}

kyun bachta hai?

Kyunki off-diagonal pairs double sum mein do baar count hote hain; half us correction ke liye hai.