Constraints — holonomic vs non-holonomic, rheonomic vs scleronomic: Question bank

2.1.1 · D5 · Physics › Analytical Mechanics › Constraints — holonomic vs non-holonomic, rheonomic vs scler

Shuru karne se pehle, poori classification ko visible kar lete hain taaki har item apne aap samajh aaye.

Recall Do sawaal jo aap KISI BHI constraint ke baare mein poochho

Axis 1 — Holonomic? Kya main ise ek equation $f (coordinates, t) = 0$ ke roop mein likh sakta hoon jisme koi velocity bacha na ho (ya velocities integrate ho jaayein)? Agar haan → holonomic. Agar nahi → non-holonomic. "Non-holonomic" ke andar do alag cheezein hoti hain: (a) ek velocity rule jo integrate hone se mana kar de, aur (b) ek inequality — neeche wala callout dekho. Axis 2 — Scleronomic vs rheonomic? Kya clock symbol $t$ explicitly appear karta hai? Koi $t$ nahi → scleronomic (frozen). Explicit $t$ → rheonomic (moving/driven). Ye alag sawaal hain. Har constraint ko har pair se ek-ek label milta hai.

Formula 2-variable integrating factor — yeh hamesha kyun exist karta hai, aur kaise dhundhein

Claim: sirf do variables mein har Pfaffian $A (x, y) d x + B (x, y) d y = 0$ integrable hoti hai, isliye genuine non-holonomy ke liye $\geq 3$ variables chahiye. Kyun: relation $A d x + B d y = 0$ har point par ek slope $\frac{d y}{d x} = - \frac{A}{B}$ define karta hai — plane mein ek field of directions. Aise first-order ODE ke solution curves $f (x, y) = c$ ki ek family hamesha hoti hai (Picard existence), aur wahi curves integrated constraint hain. Function $μ$ jo $μ A d x + μ B d y = df$ exact banata hai woh integrating factor hai, aur use yeh satisfy karna hoga: $\frac{\partial ( μ A )}{\partial y} = \frac{\partial ( μ B )}{\partial x} .$ Kaise dhundhen (standard cases): agar $\frac{1}{B} (\frac{\partial A}{\partial y} - \frac{\partial B}{\partial x})$ sirf $x$ par depend kare, toh $μ (x) = exp \int (\dots) d x$ ; similarly sirf $y$ ke liye. Teen ya zyada variables mein aisa koi $μ$ exist karna zaroori nahi — yahi woh mathematical space hai jisme rolling rehta hai. Picture dekho:

True or false — justify karo

A pendulum of fixed length swinging in a vertical plane is holonomic.

True — bob

x^{2} + y^{2} - ℓ^{2} = 0

follow karta hai, jo ek pure coordinate equation hai, isliye yeh holonomic aur scleronomic hai (koi

t

nahi).

Har constraint jo velocities involve kare woh non-holonomic hota hai.

False — ek velocity (Pfaffian) relation non-holonomic tab hoti hai jab non-integrable ho. Agar

\sum a_{i} d q_{i} = df

kisi

f

ke liye, toh integrate karke wapas coordinate equation milo aur yeh disguise mein holonomic tha.

Ek rheonomic constraint automatically non-holonomic hota hai.

False — axes independent hain. Ek bead on a wire jo fixed

ω

par spin ho rahi ho woh

y - x tan (ω t) = 0

follow karta hai, jo ek equation hai → holonomic, phir bhi explicit

t

hai → rheonomic. Holonomic AND rheonomic.

Ek holonomic constraint hamesha independent coordinates ki sankhya ek se kam karti hai.

True har independent equality constraint ke liye:

n = 3 N - k

. Equation ek coordinate ko baaki ke terms mein solve karne deti hai.

Ek non-holonomic constraint generalized coordinates ki sankhya kam karta hai.

False — yeh velocity directions restrict karta hai, reachable configurations nahi. Ek rolling disk ko ab bhi saare

(x, y, ϕ)

chahiye; kuch eliminate nahi ho sakta.

Agar time explicitly kisi constraint mein appear kare, toh mechanical energy generally conserved nahi hoti.

True — transformation

r = r (q, t)

mein ek

\partial r / \partial t

term aa jaata hai, constraint surface work kar sakti hai, aur energy conserved nahi rehti (relevant to Kinetic Energy and the Hamiltonian).

Do variables mein ek Pfaffian velocity constraint

A d x + B d y = 0

genuinely non-holonomic ho sakta hai.

False — do variables mein ek Pfaffian hamesha ek integrating factor

μ

admit karta hai (uska direction field hamesha integral curves

f = c

rakhta hai), isliye yeh hamesha integrable hai. Genuine non-holonomy ke liye teen ya zyada coupled variables chahiye.

Ek particle jo ek smooth sphere ke upar (attached nahi) rest kar raha ho, surface par rehte waqt ek holonomic constraint ke under hota hai.

Thoda sach — surface par rehte waqt

∣ r ∣ = a

(holonomic equality), lekin physical rule inequality

∣ r ∣ \geq a

hai, jise particle chhodh sakta hai. Actual constraint ek unilateral (inequality) constraint hai, jo broad sense mein non-holonomic hai.

Scleronomic constraint ka matlab hai system kabhi move nahi karta.

False — scleronomic ka matlab hai constraint surface frozen hai (explicit

t

nahi), particle static hai yeh nahi. Ek bead freely ek fixed wire par slide karta hai; wire frozen hai, bead chalti hai.

Ek inequality constraint aur ek non-integrable rolling constraint ek hi cheez hain.

False — dono ko "non-holonomic" mein rakha jaata hai, lekin pehla unilateral hai (pura region allow karta hai, on/off ho sakta hai) aur doosra ek Pfaffian velocity restriction hai (always-active surface par directions limit karta hai). Alag mechanisms, same broad label.

Galti dhundho

y = x tan (ω t)

mein

t

hai, isliye yeh non-holonomic hai." — galti dhundho.

Do axes ko confuse kiya gaya hai. Explicit

t

ise rheonomic banata hai, non-holonomic nahi. Yeh ab bhi coordinates ki equation hai, isliye holonomic hai. Sahi label: holonomic + rheonomic.

"Rolling-disk relation

d x sin ϕ - d y cos ϕ = 0

f (x, y, ϕ) = 0

mein integrate kiya ja sakta hai, isliye yeh holonomic hai." — galti dhundho.

Yeh exactness test fail karti hai aur iska koi integrating factor nahi hai (teen coupled variables), isliye ise integrate nahi kiya ja sakta. Yeh ek genuinely non-holonomic Pfaffian constraint hai.

"Kyunki pendulum bob ki speed change hoti hai, length constraint velocity par depend karta hai aur non-holonomic hai." — galti dhundho.

Constraint

x^{2} + y^{2} - ℓ^{2} = 0

mein koi velocity bilkul nahi hai. Speed ka change motion hai, constraint nahi. Yeh holonomic rehta hai.

"Moving-wall gas holonomic hai kyunki wall ki position time ki known function hai." — galti dhundho.

Constraint ek inequality hai (molecules box ke andar rehte hain) → ek unilateral, isliye non-holonomic, constraint. Wall ki motion jaanna use sirf rheonomic banata hai; inequality ko equation nahi banata. (Yeh ek non-holonomic AND rheonomic case hai.)

A d x + B d y = 0

jisme

\partial A / \partial y \neq = \partial B / \partial x

ho, non-holonomic hai." — galti dhundho.

As written exactness fail karna kaafi nahi hai — aapko integrating factor

μ

bhi check karna hoga, jo do variables mein hamesha exist karta hai (

μ (x) = exp \int \frac{1}{B} (\partial_{y} A - \partial_{x} B) d x

jab woh group sirf

x

ka ho). Isliye yeh ab bhi holonomic hai.

"Kyunki rolling disk sideways slip nahi kar sakta, uske kuch coordinates eliminate ho jaate hain." — galti dhundho.

Non-holonomic constraints velocity directions ko kill karte hain, coordinates ko nahi. Saare teen coordinates

(x, y, ϕ)

independent aur reachable rehte hain; kuch eliminate nahi hota.

Why questions

Ek non-integrable velocity constraint directions restrict kyun karta hai lekin reachable positions nahi?

Kyunki koi aisi function

f

nahi hai jise yeh constant force kare. Koi conserved

f

nahi hone se, configuration space ka koi region forbidden nahi hai — aap phir bhi kahin bhi pahunch sakte ho, bas har direction mein instantly move nahi kar sakte (jaise ek rolling coin ko parallel-park karo).

Exactness / integrating-factor test kyun chalana zaroori hai, sirf velocity constraint ko dekhne ki bajay?

Kyunki ek Pfaffian relation disguise mein holonomic ho sakti hai: agar yeh

df

ke barabar ho (shayad

μ

se multiply karne ke baad), integrate karne par coordinate equation milegi. Sirf test batata hai ki yeh sach mein integrate hone se mana karti hai ya nahi.

Genuine non-holonomy ke liye kam se kam teen variables kyun chahiye?

Do variables mein constraint plane mein ek direction field hai, jiske hamesha integral curves

f = c

hote hain (integrating factor hamesha exist karta hai). Aapko

\geq 3

coupled variables chahiye ek aisi differential form ke liye jise koi factor exact nahi bana sakta — exactly isliye ek plane mein rolling object (teen coordinates) qualify karta hai.

Rheonomic case energy conservation kyun threaten karta hai lekin scleronomic case nahi?

Ek frozen surface (

f (r) = 0

) koi work nahi karta; uska constraint force allowed motion ke perpendicular hota hai. Ek moving surface (

f (r, t) = 0

)

\partial r / \partial t

ke zariye motion ke saath push kar sakta hai, energy inject ya remove kar sakta hai.

Constraints hamen unknown reaction forces solve karne se kyun bachne dete hain?

Har holonomic equation ek unknown force ko ek geometric equation se replace karti hai, aur constraint force koi virtual work nahi karta — exactly yahi D'Alembert's Principle aur Virtual Displacements exploit karte hain unhe eliminate karne ke liye.

Non-holonomic constraints coordinate count us tarah simply reduce kyun nahi kar sakte jaise holonomic karte hain?

Kyunki unse kabhi ek solvable equation

f = 0

nahi milti jisse ek coordinate substitute kiya ja sake. Balki inhe equations of motion ke saath-saath carry kiya jaata hai, typically Lagrange Multipliers ke zariye.

Bead-on-spinning-wire example kyun standard illustration hai ki do axes independent hain?

Yeh simultaneously holonomic hai (ek clean equation

y - x tan ω t = 0

) aur rheonomic bhi (explicit

t

), jo prove karta hai ki ek constraint har pair se independently ek-ek label rakh sakta hai.

Ek coin jo ek aisi floor par roll kare jo time ke function ke roop mein tilt ho rahi ho, kyun missing fourth box hai?

No-slip rolling relation ek non-integrable Pfaffian hai (non-holonomic) AUR floor ki orientation

t

ki explicit function hai (rheonomic), isliye yeh grid ke non-holonomic + rheonomic corner ko fill karta hai.

Edge cases

Ek particle jo ek smooth sphere ke upar slide kar raha ho, exactly us instant par kya constraint status hai jab woh surface chhodta hai?

Equality

∣ r ∣ = a

hold karna band ho jaati hai aur

∣ r ∣ > a

mein switch ho jaati hai; constraint effectively off ho jaata hai. Yeh on/off switching ek unilateral (inequality) constraint ki pehchan hai → broad sense mein non-holonomic.

Ek system par do constraints independent nahi hain (ek doosre ko imply karta hai). Kitne coordinates remove hote hain?

Sirf ek — degrees-of-freedom formula

n = 3 N - k

independent constraints count karta hai. Ek redundant constraint kuch extra remove nahi karta.

Ek wire angular speed

ω

par rotate ho rahi hai, phir motor band ho jaata hai toh

ω = 0

. Constraint ke baare mein kya change hota hai?

Drive hote waqt yeh rheonomic hai (

y - x tan ω t = 0

); ek baar frozen hone par angle constant ho jaata hai aur

t

gayab ho jaata hai, isliye yeh scleronomic ban jaata hai. Holonomic label throughout unchanged rehta hai.

Kya constraint

f (r) = 0

jisme

f

identically zero ho (vacuously true) ek real constraint hai?

Nahi — ek constraint ko genuinely configurations restrict karna chahiye. Ek vacuous equation koi coordinate remove nahi karti aur kuch impose nahi karti;

k

sirf effective, independent restrictions count karta hai.

Kya ek single physical setup mein ek saath holonomic aur non-holonomic dono constraints ho sakte hain?

Haan — jaise ek disk jo ek fixed horizontal table par roll kare usmein holonomic contact condition hai (

z = R

, table par rehna) plus non-holonomic no-slip rolling relations. Har ek alag classify hota hai.

Agar aap ek bhi rheonomic constraint add karo toh scleronomic "energy = Hamiltonian" guarantee ka kya hota hai?

Yeh toot jaata hai — explicit

t

kinetic energy mein velocity-linear aur constant terms add karta hai, isliye Hamiltonian generally total mechanical energy ke barabar nahi rehta (dekho Kinetic Energy and the Hamiltonian).

Ek constraint velocities ke saath likha hai lekin algebra ke baad sab cancel ho jaate hain,

g (r, t) = 0

bachta hai. Holonomic hai ya nahi?

Holonomic — agar velocities disappear ho jaayein (ya integrate ho jaayein) aur ek pure coordinate-and-time equation bache, toh constraint configuration restrict karta hai, jo holonomic ki definition hai.

Grid ke charon boxes mein se har ek ka ek concrete member do.

Holonomic+scleronomic: rigid rod. Holonomic+rheonomic: bead on spinning wire. Non-holonomic+scleronomic: rolling disk on a fixed floor. Non-holonomic+rheonomic: coin rolling on a floor tilted as a function of time.