Foundations — Generalized coordinates — choosing them, degrees of freedom

2.1.2 · D1 · Physics › Analytical Mechanics › Generalized coordinates — choosing them, degrees of freedom

Degrees of freedom count karne se pehle, tumhe har woh notation fluently samajhna zaroori hai jo parent note use karta hai. Yeh page assume karta hai ki tum kuch nahi jaante aur har symbol ko ek picture se build karta hai. Upar se neeche padho — har block sirf woh symbols use karta hai jo uske upar define ho chuke hain.

0 — "System" aur "configuration" hote kya hain

Figure dekho. Left mein, ek bead ek room mein kahi baith hai. Yeh batane ke liye ki woh kahan hai, tum teen directions mein point karte ho. Right mein, bilkul wahi idea lekin bead ek wire par threaded hai — room ka zyaadatar hissa ab forbidden hai, aur sirf wire par wali positions real configurations hain. Poora topic usi forbidden grey region ko phenko ke baare mein hai.

1 — Ek point, aur har direction ke peeche number line

Numbers kyun? Kyunki "kahin udhar" calculate nahi ho sakta. Ek number add, subtract, aur plot ho sakta hai. Aisi number ke liye hum usually $x$, $y$, ya $z$ letter use karte hain.

2 — Position vector $\mathbf{r}$

Bundle kyun karte hain? Kyunki bar bar $x$, $y$, $z$ alag alag likhna noisy hai. Jab parent note ${\mathbf{r}}_i$ likhta hai, to bold letter keh raha hai "ek particle ke teeno numbers, packaged." Subscript $i$ agle hain.

Picture: $N$ alag alag arrows, har particle ke liye ek, har ek ka apna tag ${\mathbf{r}}_1,{\mathbf{r}}_2,\dots ,{\mathbf{r}}_N$.

3 — Raw numbers count karna: $3N$

Parent note kabhi kabhi $2N$ kyun use karta hai? Kyunki agar poora problem ek flat plane mein hai (ek pendulum ek vertical sheet mein swing kar raha hai), to "page ke bahar upar" direction kabhi change nahi hoti — woh koi number contribute nahi karti. Ek frozen direction drop karna puri game ka pehla taste hai.

4 — Functions: notation ${\mathbf{r}}_i(q_1,\dots ,q_n,t)$

Picture: ek box jisme aage dials lagi hain. Dials $(q_1,q_2,\dots )$ ghuma aur ek marble chute se neeche aayega — woh marble position ${\mathbf{r}}_i$ hai.

To jab parent likhta hai ${\mathbf{r}}_i={\mathbf{r}}_i(q_1,q_2,\dots ,q_n,t)$ woh keh raha hai: "mujhe generalized coordinates $q_1\dots q_n$ ki values do (aur clock reading $t$), aur main tumhe bata dunga ki particle $i$ actually kahan hai." Dials $q$ honest coordinates hain; box wire ya rod ki geometry chhupa leta hai.

5 — Angles as coordinates: $\theta$, $\varphi$, sin, cos

Parent note angles kyun pasand karta hai? Kyunki ek rigid rod par laga pendulum bob apni pivot se distance change nahi kar sakta — sirf apni direction change kar sakta hai. Ek number jo genuinely free hai woh tilt angle hai. $\theta$ ko coordinate use karna "fixed length" rule ko coordinate mein hi build kar deta hai.

Figure dekho: har angle $\theta$ ke liye jo tum dial kar sako, tip pivot se exactly $\ell$ distance par rehti hai — Pythagoras deta hai $x^2+y^2={\ell }^2{\sin }^2\theta +{\ell }^2{\cos }^2\theta ={\ell }^2$. Isi liye $\theta$ par switch karne par length constraint gayab ho jaati hai: woh sine aur cosine ki shape se automatically satisfy hoti hai.

6 — Equation $f(\dots )=0$ aur word "constraint"

"$=0$" kyun? Koi bhi rule "left side $=$ right side" ko "(left minus right) $=0$" form mein dhakel sakte hain. To $x^2+y^2={\ell }^2$ ban jaata hai $f=x^2+y^2-{\ell }^2=0$. Ise "something $=0$" likhne se har constraint ko same shape milti hai, jo hume unhe uniformly count karne deta hai.

7 — Symbols milake: $n=3N-k$

Ab headline formula mein har symbol define ho chuka hai. Ise English sentence ki tarah padho:

Picture: $3N$ empty boxes se shuru karo, har independent constraint ke liye ek box cross out karo, jo bachta hai use count karo. Woh survivor count $n$ hai.

8 — Prerequisite map

Solid arrows is page ka build order hain. Dotted arrows batate hain aage kahan jaana hai: Constraints — holonomic vs non-holonomic "$f=0$" idea ko aur gehraai deta hai, aur Configuration space and phase space bache hue $n$ numbers ko ek ghar deta hai. Wahan se Lagrangian mechanics — the Lagrangian L = T - V aur Euler–Lagrange equations ki machinery shuru hoti hai, orientation angles Rigid body kinematics — Euler angles se handle hote hain.

Equipment checklist

Khud test karo — right side dhako aur zor se jawab do.