Foundations — Hamilton-Jacobi equation

2.1.17 · D1 · Physics › Analytical Mechanics › Hamilton-Jacobi equation

Yeh page woh har symbol build karta hai jis par parent note rely karta hai, ek smart-12-year-old level se shuru karke. Isko Hamilton-Jacobi equation proper se pehle padho. Har entry deti hai plain meaning → picture → topic ko yeh kyun chahiye.

1. Position $q$ aur momentum $p$ — woh do numbers jo ek system ko poori tarah describe karte hain

Picture. Ek flat sheet banao. Horizontal axis $q$ hai (kahan), vertical axis $p$ hai (kitni tezi se, mass-weighted). Is sheet par ek dot = system ka ek poora snapshot. Is sheet ko phase space kehte hain.

Topic ko yeh kyun chahiye. Hamilton-Jacobi poori tarah phase space mein rehta hai. Yeh pair $(q,p)$ ko ek naye pair $(Q,P)$ mein transform karta hai. Toh pehle humein comfortable hona chahiye ki ek point — koi curve nahi — state hai, aur motion woh point hai jo iddhar-uddhar slide karta rehta hai. (Aur zyada Hamilton's Equations mein.)

$n$ moving parts ke liye hamare paas $q_1,\dots ,q_n$ aur $p_1,\dots ,p_n$ hote hain. Subscript $i$ bas yeh label karta hai ki kaun sa part hai. Parent ka ${\sum }_i$ ("sum over $i$") ka matlab hai "har part ka contribution jod lo." Jab ek system seedhi line mein chalta hai toh aksar hum us ek coordinate ko $q$ ki jagah $x$ kehte hain — same idea, alag letter.

2. Dot: $\dot{q}$ ka matlab hai "rate of change"

Picture. Phase-space sheet par, $\dot{q}$ aur $\dot{p}$ dot ke saath laga arrow of motion hain — direction aur speed jis taraf state drift kar raha hai. Agar $\dot{q}=0$ aur $\dot{p}=0$, toh dot frozen hai.

3. Derivative $\frac{d}{dt}$ aur partial derivative $\frac{\partial }{\partial q}$ — DO kinds kyun hain

Ek derivative yeh jawab deta hai: "agar main input ko thoda sa nudge karun, toh output kitna badlega?" Geometrically yeh ek curve ki slope hai.

Picture. Ek pahadi ki taraf socho jis ki height $f(q,t)$ hai.

Partial $\partial f/\partial q$ woh steepness hai agar tum bilkul east ki taraf chalo (sirf $q$ badlta hai). Total $df/dt$ woh steepness hai tumhare actual path ke saath jo dono directions mein ek saath curve karti hai.

4. Lagrangian $L$ aur action — "ek path ki cost"

Formula se pehle, energy ke do words.

Picture. Space mein time ke saath khicha hua ek path socho. Har instant par uska kuch $L$ value hota hai. Un saari values ko side by side rakho aur neeche ka area shade karo — woh shaded area us path ke liye action hai. Asli motion woh path hai jiska action stationary ho (dekho Action and Hamilton's Principle).

Topic ko yeh kyun chahiye. Parent ka punchline yeh hai ki Hamilton's principal function action hai: $S=\int Ldt+\text{const}$. Toh $S$ koi abstract fudge nahi — yeh asli trajectory ke saath accumulated "cost" hai. Jaanna ki $L$ aur action ka matlab kya hai, us identity ko magical ki jagah inevitable feel karata hai.

5. Hamiltonian $H$ — total energy $(q,p)$ ka function

Picture. Phase-space sheet ke upar, ek aisa landscape socho jis ki height har point $(q,p)$ par energy $H$ ho. Fixed energy $E$ wala system ek hiker ki tarah hai jo ek contour line $H=E$ par chhalna majboor hai.

Topic ko yeh kyun chahiye. $H$ HJ equation ka engine hai: tum Hamiltonian lete ho aur jahan bhi momentum $p_i$ dikhta hai, usse derivative $\partial S/\partial q_i$ se replace kar dete ho. $H$ ko "energy in $(q,p)$ language" samajhna hi woh substitution kuch meaningful banata hai.

6. Legendre transform — bridge $L\leftrightarrow H$

Pehle, momentum Lagrangian world mein kahan se aata hai:

Topic ko yeh kyun chahiye. Parent ke §3 mein, chain $\sum p_i{\dot{q}}_i-H=L$ bilkul yahi relation hai ulta chalaaya gaya. Yeh woh hinge hai jo "$dS/dt$ equals $\sum p\dot{q}-H$" ko clean statement "$dS/dt=L$" mein badalta hai. Tumhe isko yahan re-derive karne ki zaroorat nahi — bas shape $p\dot{q}-H=L$ ko pehchano jab woh appear ho, yeh jaante hue ki $p_i=\partial L/\partial {\dot{q}}_i$ hi woh cheez hai jo isko true banati hai.

7. Canonical transformation aur generating function

Picture. Phase-space sheet lo aur grid lines repaint karo: naye axes $Q$ aur $P$ purane ke upar khiiche gaye. Ek canonical transformation ek khaas repainting hai jo underlying flow rules ko distort nahi karta — jaise Cartesian se polar graph paper par jaana bina geometry todhe.

Type 2 kyun? Hum chahte hain ki naye momenta $P$ constants hon. Ek type-2 function pehle se $P$ ko independent input treat karta hai, isliye yeh constant new momenta design karne ka natural tool hai. (Full menagerie Canonical Transformations mein.)

8. Naya Hamiltonian $K$ (woh "Kamiltonian")

9. Separation of variables aur "complete integral" constants

Pehle, woh constant jo tab appear hota hai jab energy conserved hoti hai:

Picture. $-Et$ ko $S$ se peelna aise hai jaise ek receipt ko ek fixed "per-second rate $E$" times elapsed time, plus ek purely spatial remainder $W(q)$ mein separate karna. Tab hi possible hai jab energy rate $E$ constant ho — matlab energy conserved.

Topic ko yeh kyun chahiye. Yeh constants leftover nahi hain — yeh naye momenta $P$ hain jo poori canonical transformation ko exist karate hain. Inke bina koi map nahi hai. (Yeh baad mein Action-Angle Variables ban jaata hai.)

Prerequisite map

Left side par jo kuch bhi hai woh YEH page build karta hai; akela node Hamilton-Jacobi equation woh parent topic hai jis mein yeh sab jaata hai. Wave mechanics ke saath bridge bhi dekho Schrodinger Equation mein.

Equipment checklist

Right side chhupao aur apne aap ko test karo. Tum ready ho jab har ek instant ho.