Constraints — holonomic vs non-holonomic, rheonomic vs scleronomic: Worked examples

2.1.1 · D3 · Physics › Analytical Mechanics › Constraints — holonomic vs non-holonomic, rheonomic vs scler

Shuru karne se pehle: do words jo hum har line par use karenge, plus ek letter.

Definition Do yes/no questions (recap taaki kuch assume na ho)

Har constraint ko do independent questions ke along classify kiya jaata hai — hum unhe Axis 1 aur Axis 2 bolenge aur yahi naam har worked example mein use karenge.

Axis 1 — "kahan ya kis direction mein?"
- Holonomic ::: rule ko sirf positions aur time ki ek equation mein squeeze kiya ja sakta hai, likha jaata hai $f (r, t) = 0$ . Yahan $f$ sirf ek function hai — ek machine jo coordinates $r$ aur time $t$ khata hai aur ek single number ugalta hai; rule ye hai ki ye number $0$ hona chahiye. Ye batata hai tum kahan ho sakte ho.
- Non-holonomic ::: ise aise kisi $f (r, t) = 0$ mein squeeze nahi kiya ja sakta — ye ya toh inequality hai, ya ek velocity relation jo integrate nahi hoti. Ye batata hai tum kis direction mein move kar sakte ho, na ki tum kahan pahunch sakte ho.
Axis 2 — "kya time dikhta hai?"
- Scleronomic ::: letter $t$ (time) apne aap appear nahi karta. Allowed region frozen hai.
- Rheonomic ::: $t$ explicitly appear karta hai. Allowed region clock tick karne ke saath hilta hai.
Is page par letter == $N$ == hamesha system mein particles ki sankhya ko denote karta hai. Ordinary space mein ek single particle ko locate karne ke liye 3 numbers chahiye, isliye $N$ particles ko koi bhi constraint apply karne se pehle $3 N$ Cartesian coordinates chahiye.

Har constraint ko Axis 1 se ek label aur Axis 2 se ek label milta hai — dono axes independent questions hain.

Scenario matrix

Har row ek tarah ka constraint hai jo tumhe diya ja sakta hai. Aakhri column us example ka naam deta hai jo ise nail karta hai. "Holo?" (Axis 1) aur "Time?" (Axis 2) dono columns hamesha ek definite label se bhare hote hain. Do entries ko ek word of warning chahiye:

"test decides" (row D): ek raw velocity/differential relation ka abhi koi label nahi — tumhe integrability test (Ex 4, Ex 5) run karna hoga Holonomic ya Non-holonomic likhne se pehle.
"depends on the input" (row G) aur "mixed" (row I): ye naye labels nahi hain — inka matlab hai is row mein kaafi sub-cases bundle hain, aur specific example har sub-case ke liye ek definite Holonomic/Non-holonomic aur Scleronomic/Rheonomic label spell out karta hai. Koi bhi constraint kareebi nazar daalne ke baad "depends" nahi rehta.

Cell	Paper par dikhne wala fingerprint	Holo? (Axis 1)	Time? (Axis 2)	Worked in
A	Fixed distance / rigid link, $=$ sign, koi $t$ nahi	Holonomic	Scleronomic	Ex 1
B	$t$ wali equation (moving wall, spun wire)	Holonomic	Rheonomic	Ex 2
C	Ek inequality ( $\geq, \leq$ )	Non-holonomic	Scleronomic	Ex 3
D	Velocity/differential relation — integrability test run karna hoga	test decides	Scleronomic	Ex 4, Ex 5
E	Velocity relation jo test pass karti hai (holonomic in disguise)	Holonomic	Scleronomic	Ex 5
F	Velocity relation jo test fail karti hai (genuine rolling)	Non-holonomic	Scleronomic	Ex 4
G	Degenerate / limiting input (zero length, $ω \to 0$ , sphere)	depends on the input	depends on the input	Ex 6
H	Real-world word problem (gas in a moving piston)	Non-holonomic	Rheonomic	Ex 7
I	Exam twist — degrees of freedom sahi se count karo	mixed (see Ex 8)	mixed (see Ex 8)	Ex 8

Ab hum har cell pe kaam karte hain. Har baar fingerprint dekho.

Example 1 — Cell A · rigid dumbbell (holonomic, scleronomic)

Forecast: aage padhne se pehle label guess karo. Fixed length, koi clock nahi — kaun se do words?

"Rigid" ka matlab equation ke roop mein likho. Masses ke beech ki distance kabhi change nahi ho sakti: $∣ r_{1} - r_{2} ∣^{2} - ℓ^{2} = 0.$ Ye step kyun? "Rigid rod" ek word hai; mechanics ko ek number relation chahiye. Distance ko square karne se square root avoid hoti hai (clean hai, same information), aur ise constant ke barabar set karna exactly $f (r) = 0$ form hai.
Axis 1 test karo (holonomic?). Ye already sirf coordinates mein ek clean equation hai. ✔ Holonomic. Ye step kyun? Axis 1 poochhta hai "kya main ise $f (r, t) = 0$ ke roop mein likh sakta hoon?" — aur hamare paas exactly wahi form hai.
Axis 2 test karo (time?). Koi letter $t$ appear nahi karta. ✔ Scleronomic. Ye step kyun? Axis 2 poochhta hai "kya $t$ apne aap dikhta hai?" — yahan nahi dikhta, isliye allowed region frozen hai.
Khoye hue freedom count karo. Ek independent equality constraint exactly ek coordinate remove karta hai. Yahan $N = 2$ particles hain, isliye constraints se pehle $3 N = 6$ coordinates hain; dumbbell ke $6 - 1 = 5$ hain. Ye step kyun? Ye parent ke rule $n = 3 N - k$ se match karta hai jahan $k = 1$ hai.

Example 2 — Cell B · bead on a spinning wire (holonomic, rheonomic)

Forecast: wire ek moving line hai. Kya time andar chhupta hai, ya hum use scrub kar sakte hain?

Figure — Constraints — holonomic vs non-holonomic, rheonomic vs scleronomic

Figure mein origin se do wires dikhte hain: ek white dashed line $x$ -axis ke along ( $t = 0$ par wire) aur ek pink line $4 5^{\circ}$ par tilt hua (wire jab $ω t = 4 5^{\circ}$ ho jaata hai). Origin par chhota yellow arc turned angle $ω t$ mark karta hai, aur pink line par blue dot bead ki position $(x, y)$ hai.

Wire ki direction describe karo. Time $t$ ke baad wire $ω t$ angle se turn ho chuka hai (angle = rate $\times$ time) — figure mein ye yellow arc hai. Origin se $ω t$ angle par ek line ka slope $tan (ω t)$ hai — us line ka "rise over run". Ye step kyun? Hume wire ki tilt chahiye, aur slope exactly woh number hai jo tilt encode karta hai: $tan (angle) = \frac{vertical rise}{horizontal run}$ .
Bead ko us line par force karo. Blue dot wire par sit karna chahiye, isliye us par koi bhi point $(x, y)$ obey karta hai $\frac{y}{x} = tan (ω t)$ , yaani $y - x tan (ω t) = 0.$ Ye step kyun? Ye coordinates aur $t$ mein ek equation hai — holonomic form. Isliye moving-line word-picture genuinely holonomic hai.
Classify karo. Ye $f (x, y, t) = 0$ fit karta hai → holonomic (Axis 1). Lekin $t$ , $tan (ω t)$ ke andar baitha hai aur remove nahi kiya ja sakta (wire sach mein move karta hai) → rheonomic (Axis 2). Ye step kyun? Axis 2 purely "kya $t$ ineliminably present hai?" se decide hota hai — yahan haan.
Special instant $t = \frac{π}{4 ω}$ evaluate karo. Tab $ω t = π /4$ aur $tan (π /4) = 1$ , isliye constraint ban jaata hai $y - x = 0$ , $4 5^{\circ}$ line — ye exactly wahi pink line hai jo figure mein draw hai. Ye step kyun? Value plug karne se abstract rule ek concrete picture ban jaati hai, confirm karta hai ki wire us moment par jahan expect kiya wahan hai.

Example 3 — Cell C · particle outside a dome (non-holonomic, scleronomic)

Forecast: touch karte waqt, kya ye " $=$ " hai ya " $\geq$ "? Kaun sa coordinate remove karne se mana karta hai?

Figure mein dome ko ek blue semicircle $∣ r ∣ = a$ ke roop mein draw kiya hai; ek yellow dot us par baitha hai (particle jab contact mein hai), centre se us dot tak ek white dotted radius hai; aur ek pink dashed curve dome se peeling away dikhata hai free-flight path jab particle nikal jaata hai, region $∣ r ∣ > a$ mein.

Dome par. Contact ka matlab centre se distance radius ke barabar ho: $∣ r ∣ = a ⟹ ∣ r ∣ - 1 = 0.$ Ye step kyun? Touch karna = "distance equals radius" — ye yellow dot exactly blue semicircle par sit karta hai, equality picture.
Nikal jaane ke baad. Jab ye fly off karta hai, ye bahar hai: $∣ r ∣ > a$ — pink dashed curve circle se bahar jaati hai. Dono phases cover karne wala single rule inequality hai $∣ r ∣ \geq a .$ Ye step kyun? Particle ko dome sirf bahar push kar sakti hai, andar nahi kheench sakti — ek one-sided (normal) reaction. One-sided pushes hamesha inequalities dete hain.
Classify karo. Inequality ko har time ke liye coordinate-fixing equation mein convert nahi kiya ja sakta (ye switch off ho jaati hai jab particle nikal jaata hai). → Non-holonomic (Axis 1). Koi $t$ appear nahi karta → scleronomic (Axis 2). Ye step kyun? Axis 1 (holonomic) ek genuine equality maangta hai jo hamesha hold kare; inequality is test mein fail hoti hai.

Example 4 — Cell D & F · rolling disk (non-holonomic, scleronomic)

Forecast: ek differential mein teen ya zyada coupled variables — kya integrability test pass ya fail hoga?

Figure mein ek wavy white dashed curve hai — contact point $(x, y)$ ka traced path — jisme disk ek blue thin ellipse (almost edge-on dekha circle) us path par baitha hua hai. Ek pink arrow us direction mein point karta hai jis taraf disk face kar raha hai (uska heading $ϕ$ ), aur yellow caption woh rolling relation carry karta hai jise hum test karne wale hain.

Roll $d θ$ eliminate karo. Do rolling equations divide karo (ya cross-multiply karo): $d x sin ϕ = d y cos ϕ$ , yaani $d x sin ϕ - d y cos ϕ = 0.$ Ye step kyun? $θ$ disk ka apna spin hai; $(x, y, ϕ)$ world mein path ki shape decide karti hai holonomy, isliye hum $θ$ hide karte hain.
Ise Pfaffian $A d x + B d y + C d ϕ = 0$ naam do jahan $A = sin ϕ$ , $B = - cos ϕ$ , $C = 0$ . Ye step kyun? Exactness test (parent, "Integrability test") coefficients $A, B, C$ ke liye state kiya gaya hai; unhe naam dene se hum ise mechanically apply kar sakte hain.
Exactness test run karo. Ek genuine 3-variable Pfaffian ki integrability ke liye hume vector $a = (A, B, C)$ ko Frobenius condition satisfy karna hoga $a \cdot (\nabla \times a) = 0, a = (sin ϕ, - cos ϕ, 0),$ jahan variables $(x, y, ϕ)$ hain. Curl compute karo. Uske teen components hain $(\nabla \times a)_{x} = \partial_{y} C - \partial_{ϕ} B = 0 - sin ϕ = - sin ϕ,$ $(\nabla \times a)_{y} = \partial_{ϕ} A - \partial_{x} C = cos ϕ - 0 = cos ϕ,$ $(\nabla \times a)_{ϕ} = \partial_{x} B - \partial_{y} A = 0 - 0 = 0,$ isliye $\nabla \times a = (- sin ϕ, cos ϕ, 0)$ . Tab $a \cdot (\nabla \times a) = sin ϕ (- sin ϕ) + (- cos ϕ) cos ϕ + 0 = - sin^{2} ϕ - cos^{2} ϕ = - 1.$ Ye step kyun? Teen variables mein ek single exactness pair kaafi nahi hota; Frobenius scalar $a \cdot (\nabla \times a)$ honest integrability detector hai. Agar ye identically zero nahi hai, toh koi integrating factor exist nahi karta.
Verdict padho. Scalar $- 1$ ke barabar hai har jagah — ek non-zero constant, kisi bhi $ϕ$ ke liye kabhi zero nahi. Isliye relation kisi bhi $f (x, y, ϕ) = 0$ mein integrate nahi ho sakta. → Non-holonomic (Axis 1). Koi $t$ nahi → scleronomic (Axis 2). Ye step kyun? Non-zero Frobenius scalar exactly parent ka "no integrating factor exists (needs ≥3 variables)" statement hai, numeric banaya gaya.

Example 5 — Cell D & E · ek velocity relation jo holonomic in disguise hai

Forecast: ye lagta hai velocity constraint — lekin kya ye integrate hoti hai? Step 3 se pehle guess karo.

Coefficients naam do. $A d x + B d y = 0$ jahan $A = y$ , $B = x$ . Ye step kyun? Ex 4 ki same machinery — test karne se pehle $A, B$ lo.
Do variables hain, isliye plane exactness test use karo $\frac{\partial A}{\partial y} = \frac{\partial B}{\partial x}$ . $\frac{\partial A}{\partial y} = \frac{\partial ( y )}{\partial y} = 1, \frac{\partial B}{\partial x} = \frac{\partial ( x )}{\partial x} = 1.$ Ye equal hain → exact → integrable. Ye step kyun? Do variables mein integrating factor hamesha exist karta hai, isliye koi bhi 2-variable Pfaffian holonomic hai — ye example parent ka "genuine non-holonomy needs ≥3 variables" doosri side se dikhata hai.
Coordinate equation mein integrate karo. Ek function jiska differential $y d x + x d y$ ho, woh hai $f = x y$ (kyunki $d (x y) = y d x + x d y$ ). Isliye constraint hai $x y = const .$ → Holonomic (Axis 1). Ye particle ko ek hyperbola par fix karta hai. Koi $t$ appear nahi karta → scleronomic (Axis 2). Ye step kyun? $y d x + x d y$ ko product rule $d (x y)$ ke roop mein recognize karna hi integration hai — koi integrating factor bhi nahi chahiye.

Example 6 — Cell G · degenerate & limiting inputs

Forecast: inme se kaunsa secretly limit mein apna "moving" ya "non-holonomic" character khota hai?

(a) $ℓ \to 0$ . Constraint $∣ r_{1} - r_{2} ∣^{2} - ℓ^{2} = 0$ ban jaata hai $∣ r_{1} - r_{2} ∣^{2} = 0$ , yaani $r_{1} = r_{2}$ . Abhi bhi coordinates mein ek equation hai, abhi bhi koi $t$ nahi → holonomic (Axis 1), scleronomic (Axis 2). Lekin ab ye do particles ko ek point mein fuse kar deta hai: woh saare 3 coordinates share karte hain, 3 constraints remove karte hain, 1 nahi. Ye step kyun? Degenerate length do masses ko ek mein collapse karta hai — ek useful check ki count $n = 3 N - k$ abhi bhi kaam karta hai ( $6 - 3 = 3$ freedoms, ek single particle ke freedoms), aur labels limit ke neeche change nahi hote.
(b) $ω \to 0$ . Wire ruk jaati hai. $tan (ω t) \to tan (0) = 0$ , isliye $y - x tan (ω t) = 0$ ban jaata hai $y = 0$ , fixed $x$ -axis. Time gayab ho gaya → constraint ab scleronomic hai (Axis 2), aur abhi bhi holonomic (Axis 1). Ye step kyun? Axis 2 ek limit mein flip kar sakta hai: driving rate $ω$ kill karo aur moving surface freeze ho jaata hai, ek rheonomic constraint ko scleronomic banata hai — dikhata hai ki "depends on the input" ek genuine warning tha.
(c) $∣ r ∣ = a$ (is baar equality). Ex 3 ki inequality ke unlike, ye ek genuine equality hai jo har time hold karti hai → holonomic (Axis 1), aur koi $t$ appear nahi karta → scleronomic (Axis 2). Sphere par glued particle ke isliye $3 - 1 = 2$ degrees of freedom hain (do angles $θ, φ$ ). Ye step kyun? Ye single symbol — equality vs inequality — isolate karta hai jo Axis 1 decide karta hai, saari geometry fixed rakhke: ≥ ko = se swap karo aur wahi dome non-holonomic (Ex 3) se holonomic flip ho jaata hai.

Example 7 — Cell H · real-world word problem (gas in a moving piston)

Forecast: yahan do features ladte hain — ek wall (inequality) aur ek moving wall (time). Dono labels predict karo.

Molecule walls ke beech rehta hai. $0 \leq x \leq L_{0} + v t$ . Ye step kyun? Walls sirf andar push karti hain — one-sided pushes → inequalities, jaise Ex 3 mein.
Inequality classify karo. Inequality coordinate fix nahi kar sakti → non-holonomic (Axis 1). Ye step kyun? Dome ki tarah same reasoning: ≤ equality nahi hai.
Kya time appear karta hai? Right wall $x = L_{0} + v t$ hai — boundary $t$ ke saath move karti hai → rheonomic (Axis 2). Ye step kyun? Moving piston literally gas par kaam karta hai (aise hi gas compress/expand hoti hai), isliye enclosed molecules ki energy conserved nahi — rheonomic signature.
Verdict: har molecule ki confinement non-holonomic (Axis 1) AUR rheonomic (Axis 2) hai. Ye step kyun? Ye ek combined-label cell hai; dono axes simultaneously non-trivial hain.

Example 8 — Cell I · exam twist (degrees of freedom count karo)

Forecast: har type ka constraint kya subtract karta hai, aur kis cheez se?

Coordinate count se holonomic constraints subtract karo. $N = 3$ ke saath hum $3 N = 9$ se shuru karte hain. Do rigid rods → $k = 2$ holonomic equalities → coordinates 2 se drop: $n = 3 N - k = 9 - 2 = 7.$ Ye step kyun? Sirf holonomic (Axis 1) equalities generalized coordinates remove karti hain (parent rule $n = 3 N - k$ ).
Rolling constraint ko $n$ se subtract mat karo. Non-holonomic constraints accessible velocity directions remove karte hain, coordinates nahi. Generalized coordinates ki sankhya 7 rehti hai. Ye step kyun? Ye parent ki teesri common mistake hai — ek non-holonomic constraint configuration count reduce nahi karta.
Velocity restriction ko alag se account karo. Single rolling relation 1 instantaneous velocity direction remove karta hai, isliye system ke paas 7 coordinates hain lekin har instant par sirf $7 - 1 = 6$ independent velocity directions hain. Ye step kyun? Rolling $\overset{x}{˙}, \overset{y}{˙}, \dot{ϕ}$ ko har moment tie karta hai (Ex 4) bina reachable set ko chhote kiye — Lagrangian formalism mein ek coordinate eliminate karne ki bajay ek Lagrange multiplier se handle kiya jaata hai. Isliye "mixed" row I ek Axis-1-holonomic count ko ek Axis-1-non-holonomic effect ke saath mix karta hai.

Recall One-line fingerprints (khud test karo)

Equality, koi $t$ nahi ::: holonomic + scleronomic (Ex 1) Equality jisme $t$ andar ho ::: holonomic + rheonomic (Ex 2) Inequality ::: non-holonomic (Ex 3, 7) Differential relation ::: integrability test run karo — kuch bhi ho sakta hai (Ex 4, 5) Frobenius scalar $a \cdot (\nabla \times a) \neq = 0$ ::: genuinely non-holonomic (Ex 4) Non-holonomic constraints subtract karte hain ::: velocity directions se, coordinate count se nahi (Ex 8)