Generalized coordinates — choosing them, degrees of freedom

2.1.2 · Physics › Analytical Mechanics

Generalized coordinate KYA hai?

Degrees of freedom KAISE count karein — formula derive karo

YE KYU KAAM KARTA HAI: Sabse zyada wasteful description se shuru karo aur redundancy subtract karo.

Unconstrained se shuru karo. $N$ point particles ko 3D mein locate karne ke liye $3 N$ Cartesian numbers chahiye. Yahi raw configuration space hai.
Har holonomic constraint ek equation hai. Ek equation $f (\dots) = 0$ tumhe ek variable ko doosron ke terms mein solve karne deti hai — to ye exactly ek independent number ko khatam kar deti hai.
Subtract karo. Agar $k$ independent holonomic constraints hain:

$n = 3 N - k$

Rigid body ke liye $6$ KYU? Ek rigid body mein infinitely many particles hote hain lekin rigidity constraints ( $∣ r_{i} - r_{j} ∣ =$ const) 6 ke siwa sab numbers ko pin kar dete hain: 3 ye batane ke liye ki body kahan hai, 3 ye batane ke liye ki woh kis orientation mein hai.

Figure — Generalized coordinates — choosing them, degrees of freedom

Achche generalized coordinates KAISE choose karein

Checklist:

Kya har $q$ configuration ko change karta hai? (independence)
Kya sab $q$ 's ko fix karne se har particle fix ho jaata hai? (completeness)
Kya natural symmetries (angles, track ke along distances) constraints ko eliminate karti hain? (convenience)

Worked Examples

Common Mistakes

Flashcards

Generalized coordinate kya hai?

Ek independent variable jo ek system ki configuration fully specify karne mein help karta hai; length hona zaruri nahi (angle, ratio, charge, etc. ho sakta hai).

Degrees of freedom define karo.

Configuration specify karne ke liye zaruri independent generalized coordinates ki minimal sankhya; holonomic systems ke liye

n = 3 N - k

Holonomic constraint kya hai?

Woh jo coordinates aur time ke beech equation

f (r_{1}, \dots, r_{N}, t) = 0

ke roop mein expressible ho.

Har independent holonomic constraint kitne DOF remove karta hai?

Exactly ek.

Ek free rigid body ke DOF?

6 (3 translation + 3 rotation).

Ek plane mein simple pendulum ke DOF?

1 (angle

θ

Pendulum ke liye

θ

achcha coordinate kyu hai?

Kyunki

x = ℓ sin θ, y = - ℓ cos θ

automatically length constraint satisfy karta hai, usse eliminate kar ke.

Rigid diatomic molecule mein 6 ki jagah 5 DOF kyu hote hain?

Bond-length constraint 1 hatata hai; bond axis ke around rotation point atoms ke liye invisible hai, to 3 translation + 2 orientation angles bachte hain.

Kya ek time-dependent (rheonomic) constraint holonomic hota hai?

Haan — ye phir bhi equation

f (r, t) = 0

hai aur ek DOF remove karta hai;

t

parameter hai, coordinate nahi.

Pure rolling DOF counting ke liye tricky kyu hai?

Ye typically non-holonomic hai (ek velocity constraint jo position equation mein integrate nahi ho sakta), to

n = 3 N - k

apply nahi ho sakta.

Recall Feynman: ek 12-saal ke bachhe ko explain karo

Socho ek circular track par ek toy train hai. Train ek bade kamre mein hai, to tum use describe kar sakte ho ki woh kitni left hai, kitni forward hai, kitni upar hai — teen numbers. Lekin ye bakwaas hai! Train sirf loop ke around ja sakti hai, to actually ek number — "track par kitni door hai" — sab kuch bata deta hai. Woh single number uska generalized coordinate hai, aur "ek" uska degrees of freedom hai. Jab bhi koi cheez kisi path ya surface par stuck ho, tum useless numbers phenko aur sirf woh rakho jo actually change hote hain. Tum kitne rakhte ho ye count karna = degrees of freedom count karna.

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