Generalized coordinates — choosing them, degrees of freedom: Question bank

2.1.2 · D5 · Physics › Analytical Mechanics › Generalized coordinates — choosing them, degrees of freedom

True or false — justify karo

A generalized coordinate ka unit length mein hona zaroori hai.

False. Coordinate sirf independent aur complete hona chahiye — ek angle (radians), ek ratio, ya phir charge bhi qualify karta hai. Dekho Configuration space and phase space jahan axes aksar mixed dimensions rakhte hain.

Har holonomic constraint exactly ek degree of freedom remove karta hai.

True, bas tab jab woh doosron se independent ho. Ek equation

f (\dots, t) = 0

ek variable ko baaki ke terms mein solve karne deta hai, ek number ko khatam karta hai.

Ek time-dependent (rheonomic) constraint ek degree of freedom add karta hai kyunki time vary kar sakta hai.

False. Time ek parameter hai, koi coordinate nahi jo tum freely dial kar sako; constraint

f (r, t) = 0

phir bhi ek DOF remove karta hai. Dekho Constraints — holonomic vs non-holonomic.

Teen alag constraint equations likhne se hamesha teen degrees of freedom remove hote hain.

False. Sirf independent equations count karte hain; agar ek doosron se follow hota hai toh woh kuch remove nahi karta.

Ek free rigid body space mein 6 degrees of freedom rakhta hai chahe usme kitne bhi atoms hon.

True. Rigidity saare inter-particle distances fix kar deti hai, 3 position ke liye aur 3 orientation ke liye bacha deti hai — dekho Rigid body kinematics — Euler angles.

Generalized coordinates ki sankhya hamesha

3 N - k

ke barabar hoti hai.

False in general. Woh formula sirf holonomic systems ke liye hold karta hai; non-holonomic (velocity) constraints motion restrict kar sakte hain bina position coordinates ki count ghataaye.

Ek rigid diatomic molecule jo do point atoms ki tarah model kiya gaya hai, uske 6 DOF hain.

False — uske 5 hain. Bond-length equation ek remove karta hai, aur bond axis ke baare mein ghoomna point atoms ke liye kuch nahi hiltaa, toh woh "rotation" ek DOF nahi hai.

Pendulum ke liye

(x, y)

ki jagah

θ

choose karna degrees of freedom ki sankhya badal deta hai.

False. DOF system ki property hai, coordinate choice ki nahi; dono ek 1-DOF system describe karte hain, lekin

θ

constraint ko bake in kar leta hai isliye woh cleaner hai.

Do beads ek seedhi wire par (3D mein) ke 2 degrees of freedom hain.

True. Har bead ka axis pe confinement do equations hai (

y_{i} = z_{i} = 0

), toh

k = 4

aur

n = 3 (2) - 4 = 2

Do constrained beads ke beech spring add karne se unke degrees of freedom badal jaate hain.

False. Spring ek force hai, constraint nahi; woh dynamics aur energy badalta hai lekin coordinates ki count untouched rehne deta hai.

Galti pakdo

"Ek rotating wire par bead ke 2 DOF hain kyunki uski position ke liye

r

aur wire ka angle chahiye."

Wire ka angle forced hai

ω t

hone ke liye — woh free nahi hai. Sirf

r

independent hai, toh system ka 1 DOF hai; angle time ke zariye set hota hai.

"Pendulum constraint hai

x^{2} + y^{2} = ℓ^{2}

, aur ground

y = 0

par hai, toh

k = 2

aur

n = 0

Ground ka ek swinging bob ke liye koi role nahi — koi

y = 0

constraint nahi hai. Sirf rod length constrain karta hai, toh

k = 1

aur

n = 1

"Ek line par bina fisal ke rolling disk ek DOF kho deta hai, toh

n = 3 N - k

use karo."

Pure rolling aam taur par non-holonomic hoti hai — woh

\overset{x}{˙} = R \dot{θ}

(ek velocity relation) constrain karta hai jo

f (x, θ) = 0

mein integrate nahi ho sakta. Configuration space dono

x

aur

θ

rakhta hai; dekho Constraints — holonomic vs non-holonomic.

"Ek rigid body ko

3 N

coordinates chahiye minus rigidity constraints, aur

N \to \infty

, toh uske infinitely many DOF hain."

Constraints bhi unbounded grow karte hain aur 6 ke siwa sab cancel kar dete hain. Sahi count 6 hai, jo seedhe reasoning se milta hai (position + orientation), naive subtraction se nahi.

"Teen non-collinear point atoms wale molecule ke liye,

k = 3

(teen bond lengths) toh

n = 9 - 3 = 6

Yeh actually teen general fixed distances ke liye sahi hai —

9 - 3 = 6

(3 translation + 3 rotation). Trap yeh hai ki assume kar lo ki yeh "hona chahiye" 5 jaise dumbbell; teen non-collinear atoms ka ek genuine third rotational DOF hota hai.

"Kyunki

t

x = r cos ω t

mein appear karta hai, time generalized coordinates mein se ek hai."

Time explicitly sirf isliye appear karta hai kyunki constraint rheonomic hai;

t

evolution ka independent parameter hai, koi

q

nahi jo tum fixed instant par vary karo.

Why questions

Do atoms wala dumbbell kyu third orientation angle use nahi kar sakta?

Unhe join karne wali line ke baare mein do point atoms ko rotate karne se identical configuration milti hai — "spin" angle physically invisible hai, toh woh ek independent coordinate nahi hai.

Constraint ko coordinate mein bake in karna (jaise

θ

) "smart" choice kyu maana jaata hai?

Kyunki constraint equation phir coordinate ki har value ke liye automatically hold karti hai, toh woh gayab ho jaati hai aur tum sirf truly free variables ke saath kaam karte ho — jo cleanly Lagrangian mechanics — the Lagrangian L = T - V mein feed hota hai.

Constraints subtract karne se pehle hum independence kyu check karte hain?

Ek dependent constraint koi nayi information nahi leta; use subtract karna ek restriction ko double-count karta hai aur galat (zyada chhota) DOF deta hai.

Hum term by term

3 N - k

ki jagah rigid body ke liye

6

kyu count karte hain?

Kyunki rigidity relations ki sankhya enormous aur highly redundant hoti hai; "kahan hai + kaise oriented hai" se seedha sochne par 6 independent numbers milte hain bina redundancy se jujhe.

Ek rheonomic constraint phir bhi ek DOF kyu remove karta hai jabki

r

t

par depend karta hai?

Kisi bhi fixed instant par woh abhi bhi coordinates relate karne wala ek equation hai, toh woh ek variable pin karta hai; time sirf kaun si configuration allowed hai woh shift karta hai, na ki kitni free hain.

Do spring-linked beads ke liye center-of-mass

X

aur separation

s

prefer kyu kiya jaata hai?

Spring energy sirf

s

par depend karti hai, toh equations decouple ho jaate hain —

X

freely evolve karta hai jabki

s

saari interaction carry karta hai, Euler–Lagrange equations simplify karte hue.

Non-holonomic constraint

n = 3 N - k

counting kyu tod deta hai?

Woh velocities restrict karta hai (differential relations) jo position equations mein integrate nahi ho sakti, toh woh kuch motions forbid karta hai bina koi configuration forbid kiye — position-coordinate count zyada rehta hai.

Edge cases

3D mein koi constraint nahi wale ek single free particle ka DOF kya hai?

3 — raw

(x, y, z)

; subtract karne ke liye kuch nahi,

n = 3 (1) - 0 = 3

Ek particle jo ek fixed point se pin kiya gaya ho (teeno coordinates fixed) ka DOF kya hai?

0. Teen independent constraints use fully lock karte hain:

n = 3 - 3 = 0

, ek degenerate "system" jo hil nahi sakta.

Ek bead jo 3D mein ek surface se constrained hai (ek equation

f (x, y, z) = 0

) — kitne DOF?

2. Ek holonomic equation teen mein se ek number remove karta hai, ek 2-parameter surface pe roam karne ke liye.

Agar tum

2 ℓ^{2} = 2 (x^{2} + y^{2})

ko ek "doosre" pendulum constraint ke roop mein likho?

Woh algebraically same equation hai scaled — independent nahi — toh woh kuch remove nahi karta; phir bhi

n = 1

Kya ek system ke zero generalized coordinates ho sakte hain phir bhi valid configuration ho?

Haan — ek fully constrained (rigidly fixed) system ka

n = 0

hai; uski exactly ek allowed configuration hai aur move karne ki koi freedom nahi.

Do coincident particles jo same point par force kiye gaye hain (

r_{1} = r_{2}

, teen equations) — DOF?

n = 3 (2) - 3 = 3

. Woh ek point ki tarah saath move karte hain; teen equality constraints chhe numbers ko teen mein collapse kar dete hain.

Ek pendulum jiska length

ℓ (t) = ℓ_{0} + v t

ki tarah grow karta hai — kitne DOF?

Phir bhi 1. Constraint

x^{2} + y^{2} = ℓ (t)^{2}

rheonomic lekin holonomic hai, ek DOF remove karta hai; angle

θ

sole free coordinate rehta hai.

Connections

Constraints — holonomic vs non-holonomic
Configuration space and phase space
Lagrangian mechanics — the Lagrangian L = T - V
Euler–Lagrange equations
Rigid body kinematics — Euler angles
Principle of virtual work and d'Alembert's principle