2.1.1
Intuition Badi picture (WHY constraints matter karte hain)Newton ke laws mein har force chahiye, including woh unknown forces jo particles ko tracks, surfaces, ya wires par rokti hain (rod tension, normal reaction, etc.). Ye forces irritating hoti hain: hum inhe advance mein nahi jaante. Constraints woh geometric rules hain jo un unknown forces ko replace karti hain. Yeh poochhne ki bajay ki "bead ko wire par kaun si force rokti hai?", hum kehte hain "bead ke coordinates f ( r ) = 0 independent coordinates ki sankhya ghata sakte hain aur (baad mein) constraint forces ko poori tarah eliminate kar sakte hain. Yahi Lagrangian mechanics ki poori motivation hai.
Ek constraint ek kinematic restriction hai positions par (aur possibly velocities par bhi) kisi system ki, jo coordinates { r i } { r ˙ i } t
N 3 N Har independent equality constraint ek coordinate remove karti hai.
Bacha hua number degrees of freedom n = 3 N − k k
Dono classification axes independent sawaal hain:
Holonomic vs non-holonomic — Kya constraint ko sirf coordinates aur time mein equation ke roop mein likha ja sakta hai? Rheonomic vs scleronomic — Kya time explicitly appear karta hai?
Definition Holonomic constraintEk constraint holonomic hota hai agar use is tarah express kiya ja sake:
f ( r 1 , … , r N , t ) = 0. configuration restrict karta hai (aap kahan ho sakte hain). Yeh independent coordinates ki sankhya ghataata hai.
Definition Non-holonomic constraintEk constraint non-holonomic hota hai agar use aisi equation mein reduce nahi kiya ja sake. Typical forms:
Inequalities: f ( r , t ) ≥ 0 Non-integrable velocity relations: ∑ i a i ⋅ d r i + a 0 d t = 0 f exact differential nahi hai.
Velocities par ek constraint ∑ a i d q i + a 0 d t = 0 secretly holonomic hota hai agar koi function f integrate kar sako , to f = koi integrating factor exist nahi karta , to constraint genuinely motion ki directions limit karta hai lekin reachable configurations ka set nahi — yahi non-holonomic ki pehchaan hai (jaise ek rolling coin kisi bhi position+orientation tak pahunch sakta hai, rolling restriction ke bawajood).
Definition Scleronomic vs Rheonomic
Scleronomic ("rigid time"): time explicitly appear nahi karta , f ( r ) = 0 frozen hai.Rheonomic ("flowing time"): time explicitly appear karta hai , f ( r , t ) = 0 move/change karta hai.
Intuition Distinction KYU matter karta haiEk scleronomic, holonomic system ke liye kinetic energy velocities mein ek pure quadratic hoti hai aur T = 2 1 ∑ m q ˙ 2 ω r = r ( q , t ) ∂ r / ∂ t T mechanical energy conserved nahi bhi ho sakti . Time-axis classification actually energy conservation ke baare mein ek sawaal hai.
Worked example 1 — Rigid body / dumbbellDo masses ek rigid rod se jude hue hain jiska length ℓ ∣ r 1 − r 2 ∣ 2 − ℓ 2 = 0. Yeh step kyun? Distances fixed hain → sirf coordinates mein equation → holonomic . Koi t scleronomic . Unknown rod tension ko is ek equation se replace kiya jaata hai, jo 1 DOF remove karti hai.
Worked example 2 — Bead on a rotating wireEk bead ek straight wire par constrained hai jo plane mein fixed angular speed ω y − x tan ( ω t ) = 0. Yeh step kyun? Yeh coordinates mein equation hai → holonomic , lekin t rheonomic . Wire ki externally driven motion bead par kaam kar sakti hai → energy conserved nahi.
Worked example 3 — Particle on/inside a sphere of radiusa ∣ r ∣ ≥ a (outside) . Yeh step kyun? Yeh ek inequality hai, equality nahi, isliye yeh ek coordinate fix nahi kar sakti → non-holonomic . Ek particle jo dome se slide karti hai woh ∣ r ∣ = a jab tak woh leave nahi karti , phir ∣ r ∣ > a t
Worked example 4 — Rolling disk (classic non-holonomic)Ek vertical disk plane mein bina slipping ke roll kar rahi hai, position ( x , y ) ϕ R d x = R cos ϕ d θ , d y = R sin ϕ d θ , θ Yeh step kyun? Ye x , y , ϕ , θ d θ d x sin ϕ − d y cos ϕ = 0 exact differential nahi hai aur iska koi integrating factor nahi hai (3+ coupled variables), isliye ise f ( x , y , ϕ ) = 0 nahi kiya ja sakta. → Non-holonomic, scleronomic . Disk ko kisi bhi ( x , y , ϕ )
Common mistake "Har velocity-dependent constraint non-holonomic hota hai."Kyun sahi lagta hai: Non-holonomic examples (rolling) mein hamesha velocities involve hoti hain, isliye log dono ko barabar maan lete hain.
Fix: Ek velocity constraint non-holonomic hota hai sirf tab agar non-integrable ho . Agar ∑ a i d q i = df f
Common mistake "Rheonomic matlab non-holonomic hai."Kyun sahi lagta hai: Dono "zyada complicated" constraints lagte hain.
Fix: Ye alag-alag sawaalon ke jawaab dete hain. Ek spinning wire par bead (y = x tan ω t holonomic AND rheonomic hai. Ye axes orthogonal hain — ek constraint mein har pair se ek label hota hai.
Common mistake "Constraints hamesha degrees of freedom ko unki sankhya ke hisaab se ghataate hain."Kyun sahi lagta hai: Holonomic equalities ke liye sach hai.
Fix: Non-holonomic constraints accessible velocity directions ghataate hain lekin generalized coordinates ki sankhya / reachable configurations ko nahi . Rolling disk apne saare coordinates rakhta hai.
Holonomic constraint ko kya define karta hai? Ise ek equation f ( r 1 , … , r N , t ) = 0
Holonomic constraint degrees of freedom ko kaise affect karta hai? Har independent ek ek coordinate remove karta hai: n = 3 N − k
Velocity constraint ko non-holonomic kya banata hai? Yeh non-integrable hota hai — koi f
Scleronomic vs rheonomic? Scleronomic = koi explicit t t
Rheonomic energy conservation ko kyun threaten karta hai? r = r ( q , t ) ∂ r / ∂ t
Fixed ω Holonomic (y − x tan ω t = 0
Rolling vertical disk ko classify karo. Non-holonomic (non-integrable rolling) aur scleronomic.
Box ke andar confined particle ko classify karo. Non-holonomic (inequality) aur scleronomic.
A d x + B d y = 0 Integrable tab jab ∂ A / ∂ y = ∂ B / ∂ x
Genuine non-holonomy sirf 2 variables ke saath kyun nahi ho sakti? 2 variables mein integrating factor hamesha exist karta hai, isliye aisa koi bhi Pfaffian integrable hota hai.
Recall Feynman: ek 12-saal ke bachche ko samjhaoEk toy train imagine karo. Ek holonomic rule aisa hai jaise "train ko floor par bani painted line par rehna chahiye" — tum exactly likh sakte ho ki use kahaan allowed hai. Ek non-holonomic rule aisa hai jaise "wheels sideways slide nahi kar sakte" — yeh pin down nahi karta ki train kahan pahunchegi ; train phir bhi floor par kisi bhi jagah pahunch sakti hai, bas woh wahaan pahunchne ke liye sideways slide nahi kar sakti. Scleronomic = painted line kabhi move nahi karti. Rheonomic = koi painted line ko idhar-udhar drag kar raha hai jabki train follow karne ki koshish karti hai. Do bilkul alag cheezein: ek yeh hai ki rule ka shape kya hai , doosra yeh hai ki rule time ke saath badalta hai ya nahi .
Mnemonic Dono axes yaad rakho"HoloN = whole eqN" (Holonomic → poori equation coordinates mein likh sakte ho).
"scleRO = ROck-still time, RHEo = RHEr (river) flowing time." Sclero = rock = koi t t
Generalized Coordinates — holonomic constraints independent q i Lagrangian Mechanics — holonomic constraints L ( q , q ˙ , t ) D'Alembert's Principle — constraint forces ko virtual work ke zariye handle karta hai.Lagrange Multipliers — non-holonomic / Pfaffian constraints ka standard tool.Kinetic Energy and the Hamiltonian — scleronomic ⇒ H = Virtual Displacements — rheonomic vs scleronomic mein fark yeh hai ki δ r ∂ r / ∂ t
Unknown constraint forces
Degrees of freedom n = 3N - k
Inequalities or non-integrable velocity
Exactness condition dA/dy = dB/dx
