Foundations — Generalized momenta and generalized forces

2.1.7 · D1 · Physics › Analytical Mechanics › Generalized momenta and generalized forces

Is page par kuch bhi assumed nahi hai. Agar parent note mein koi symbol likha tha, toh hum use yahan pehle ek picture se banate hain. Ise ek baar padho aur tum kabhi bhi notation par stuck nahi rahoge.

0. Story ke teen characters

Kisi bhi symbol se pehle, un teen quantities se milo jinhe poora subject juggle karta hai:

Is page ka baaki hissa right column ko define karta hai, ek symbol at a time, hamesha ek picture se joda hua.

1. Vector ${\vec{r}}_i$ — "particle number $i$ kahan hai?"

Upar chhota arrow, $\stackrel{⃗}{}$, ek promise hai: "is cheez mein length aur direction dono hain." Ek plain number (jaise temperature) ka koi direction nahi hota; ek position ka hota hai — isliye use arrow milta hai.

Topic ko yeh kyun chahiye. Physics ki shuruat "cheezein kahan hain" se hoti hai. Newton ne sab kuch in arrows ke terms mein likha. Parent note ka poora game yeh hai ki arrow likhna band karo aur uski jagah single letters $q_j$ likho — lekin pehle hume pata hona chahiye ki hum kya replace kar rahe hain.

2. Generalized coordinate $q_j$ — "har freedom ke liye ek honest number"

Ek pendulum dekho. Bob kisi $(x,y)$ par rehta hai, lekin $x$ aur $y$ free nahi hain: rod force karta hai $x^2+y^2={\ell }^2$. Toh actually sirf ek independent number hai — swing angle.

Degrees of freedom $n$ = dials ki sankhya. Constraints $n$ ko kaise kam karte hain uski poori kahani ke liye Generalized coordinates and constraints dekho.

Kyunki jab dials set ho jaate hain toh har particle ka Cartesian arrow fix ho jaata hai, hum likh sakte hain ${\vec{r}}_i={\vec{r}}_i(q_1,\dots ,q_n,t).$ Ise zor se padho: "particle $i$ ki position ek recipe hai jo dial settings (aur shayad clock time $t$) leta hai aur ek arrow return karta hai."

3. Dot: ${\dot{q}}_j$ aur $\dot{x}$ — "change ki rate"

Derivative kyun, aur sirf "difference" kyun nahi? Kyunki motion smooth aur continuous hoti hai; hum ek moment par instantaneous rate chahte hain, kisi time chunk ka average nahi. Derivative exactly woh tool hai jo "how fast, right now?" ka jawab deta hai — Euler-Lagrange Equation dekho jahan derivatives dominate karte hain.

Topic ko yeh kyun chahiye. Motion ki energy speed par depend karti hai, aur ek dial ki speed ${\dot{q}}_j$ hai. Har momentum formula mein yeh dots honge.

4. Partial derivative $\frac{\partial }{\partial q_j}$ — "sirf ek dial wiggle karo"

Yeh parent page par sabse zyada dare jaane wala symbol hai. Ek baar dekh lo toh gentle hai.

Topic ko yeh kyun chahiye — har jagah.

• $\frac{\partial {\vec{r}}_i}{\partial q_j}$ = "particle $i$ kis direction mein aur kitna slide karta hai jab main dial $j$ ko thoda sa turn karun." Yeh do coordinate worlds ke beech ka bridge arrow hai.
• $p_j=\frac{\partial L}{\partial {\dot{q}}_j}$ aur $Q_j=-\frac{\partial V}{\partial q_j}$ dono is symbol par live karte hain.

5. Dot product $\vec{A}\cdot \vec{B}$ — "do arrows kitna agree karte hain?"

Yeh operation kyun aur ordinary multiplication kyun nahi? Kyunki hum aksar ek arrow ka woh hissa chahte hain jo doosre ke saath lie karta hai — for example, ek force ka sirf woh hissa jo motion ki direction mein act karta hai work karta hai. $\cos \varphi$ factor exactly woh overlap extract karta hai:

• arrows aligned ($\varphi =0$): $\cos =1$, full agreement, sabse bada result;
• perpendicular ($\varphi =90^{\circ }$): $\cos =0$, zero — ek sideways force koi work nahi karta;
• opposed ($\varphi =180^{\circ }$): $\cos =-1$, negative — force motion se ladhti hai.

Topic ko yeh kyun chahiye. Work hai $\delta W={\sum }_i{\vec{F}}_i\cdot \delta {\vec{r}}_i$ aur generalized force hai $Q_j={\sum }_i{\vec{F}}_i\cdot \partial {\vec{r}}_i/\partial q_j$. Dono dot products hain: "push allowed motion se kitna agree karta hai?"

6. Summation sign $\sum$ — "sab cheezein add karo"

Topic ko yeh kyun chahiye. Ek system mein kai particles hain (${\sum }_i$) aur kai dials (${\sum }_j$). $\delta W={\sum }_j{Q}_j\delta q_j$ kehta hai "total work = (har generalized force) × (us dial ki wiggle) ka sum."

7. Virtual displacement $\delta {\vec{r}}_i$ — "ek imaginary, time-frozen nudge"

Topic ko yeh kyun chahiye. Isi se D'Alembert's Principle aur $Q_j$ ki definition banti hai. Yeh bhoolna ki $\delta$ time freeze karta hai (ek stray $\partial /\partial t$ rakh dena) ek classic error hai jo parent page par flag ki gayi hai.

8. Energies $T$, $V$ aur Lagrangian $L$

9. Chaar naye symbols ek saath

Ab parent page ke headline boxes plain sentences ki tarah padhte hain:

Real momentum aur torque se link Angular momentum mein spell out hai; ek cyclic coordinate apna momentum kyun conserve karta hai uski deep reason Noether's Theorem hai; aur yeh sab aage kahan jaata hai woh Hamiltonian Mechanics hai.

Prerequisite map

Equipment checklist

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