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

2.1.1 · D1 · Physics › Analytical Mechanics › Constraints — holonomic vs non-holonomic, rheonomic vs scler

Yeh D1 Foundations page hai parent topic ke liye. Hum assume karte hain ki tumne kuch bhi nahi dekha. Hum har woh symbol list karte hain jis par parent note depend karta hai, har ek ko plain-words meaning dete hain, ek picture dete hain, aur reason batate hain ki topic ko uski zaroorat kyun hai — ek aisi order mein build kiya gaya jahan har rung apne neeche wale par khada ho.

0 — Ek particle aur woh kahan rehta hai

Yeh kehne ke liye ki dot "kahan" hai, humein ek address chahiye. Flat space mein woh address teen numbers hai.

Topic ko iska zaroorat kyun hai: har constraint equation in numbers ke baare mein ek sentence hai. Kisi dot ki position restrict karne se pehle hum usse name karne mein capable hona chahiye.

1 — Teen numbers ko bundle karna: position vector $\mathbf{r}$

Har baar $(x,y,z)$ likhna clumsy hai. Hum poore address ko ek bold naam dete hain.

Topic ko iska zaroorat kyun hai: parent constraints ko $f(\mathbf{r})=0$ jaisi form mein likhta hai. Ek symbol $\mathbf{r}$ trio $(x,y,z)$ ko hide karta hai taaki formulas chhote rahein.

2 — Kai particles: subscript $i$ aur count $N$

Real systems mein ek se zyada dot hote hain: ek rod par do masses, molecules ki ek gas.

Topic ko iska zaroorat kyun hai: yeh count $3N$ numbers ka starting stockpile hai. §6 mein hum is mein se rules subtract karenge taaki pata lage ki actually kitni freedom bachi hai.

3 — Samay ke saath change: upar ka dot, $\dot{\mathbf{r}}$

Ek constraint sirf kahan ho woh restrict nahi kar sakta balki kitni tez aur kis direction mein move ho woh bhi restrict kar sakta hai. Isliye hum velocity ke liye ek symbol chahiye.

Topic ko iska zaroorat kyun hai: distinction holonomic vs non-holonomic exactly "kya rule sirf $\mathbf{r}$ ke baare mein hai, ya woh truly $\dot{\mathbf{r}}$ involve karta hai?" hai.

4 — Rule khud: function $f$ aur "$=0$"

Topic ko iska zaroorat kyun hai: holonomic (§9 mein formally defined) ka matlab exactly "rule ko $f(\mathbf{r},t)=0$ ke roop mein likha ja sakta hai" hai. Sab kuch is statement ke shape ko pehchanne par depend karta hai.

5 — Distance aur length: $|\mathbf{r}|$ aur $|{\mathbf{r}}_1-{\mathbf{r}}_2|$

Sabse common constraint ek distance fix karta hai. Hum uske liye symbol chahiye.

Topic ko iska zaroorat kyun hai: dumbbell constraint $|{\mathbf{r}}_1-{\mathbf{r}}_2{|}^2-{\ell }^2=0$ yehi idea hai. Yahan ==$\ell$== fixed rod length hai (ek single number); distance ko square karne se square root avoid hota hai aur algebra saaf rehti hai. Gour karo ki hum yeh constraint sirf abhi likh sakte the — ${\mathbf{r}}_1$, ${\mathbf{r}}_2$, magnitude bars, aur $\ell$ har ek ka meaning hone ke baad.

6 — Freedom count karna: degrees of freedom $n$, constraint count $k$

Topic ko iska zaroorat kyun hai: $3N$ ko $n$ tak shrink karna constraints study karne ka poora payoff hai; yeh seedha Generalized Coordinates mein jaata hai.

7 — Tiny changes: differential $d$ aur step-rules

Velocity rules aksar dot ki jagah $d$ ke saath likhi jaati hain. Same idea, alag dress.

Step-rule ka general form likhne se pehle, humein uske pieces ke names chahiye.

Topic ko iska zaroorat kyun hai: rolling-disk rule (§8 mein build kiya gaya) exactly is step form mein likha gaya hai precisely kyunki woh ek $f(\mathbf{r})=0$ mein collapse nahi ho sakta.

8 — Rules ke andar appear hone wale tools

Parent kuch named math tools use karta hai. Har ek ek specific question ka jawab deta hai. Hum angle-and-heading symbols yahan milte hain, pehli baar jab woh needed hain, aur hum do example systems setup karte hain jinse woh belong karte hain.

Topic ko inki zaroorat kyun hai: yeh woh only extra math tools hain jo parent invoke karta hai, aur har ek isliye choose kiya gaya kyunki woh ek precise question ka answer deta hai — slope-with-time ($\tan$), ek heading ko components mein split karna ($\cos ,\sin$), ya test karna ki kya ek step-rule ek position rule hide karta hai ($\partial$).

9 — Do central terms, aakhirkar defined

Ab har woh symbol jo woh use karte hain in place hai, isliye hum topic ki headline definitions clearly state kar sakte hain.

Prerequisite map

Har foundation topic ko feed karta hai: coordinates aur vectors humein positions name karne dete hain, function $=0$ shape humein rule likhne deti hai, velocity + differentials humein holonomic ko non-holonomic se tell karne dete hain, aur count $3N-k$ humein rules ko fewer variables ke roop mein cash in karne deta hai. Yahan se tum Generalized Coordinates aur Lagrangian Mechanics ke liye ready ho; integrability machinery Lagrange Multipliers tak jaati hai, aur "kya surface move karta hai" wala question Kinetic Energy and the Hamiltonian tak. Tiny allowed moves ke baare mein stepwise reasoning Virtual Displacements aur D'Alembert's Principle se connect hoti hai.

Equipment checklist

Aage badhne se pehle har ek ka plain meaning do — right side cover karo aur khud ko test karo.