Foundations — Hamiltonian — definition H = Σpᵢq̇ᵢ − L

2.1.11 · D1 · Physics › Analytical Mechanics › Hamiltonian — definition H = Σpᵢq̇ᵢ − L

Is page pe kuch bhi assume nahi kiya gaya. Har squiggle jo tum parent note mein miloge, woh yahan se ground up banaya gaya hai, ek aisi order mein jahan har item sirf pehle walon pe lean karta hai.

1. Ek "system" aur uski state

Socho ek bead wire pe slide kar raha hai, ya ek pendulum jhool raha hai. Ek instant mein motion ke baare mein sab kuch jaanne ke liye, tumhe do tarah ke facts chahiye:

• Kahan hai? — ek position.
• Kaise chal raha hai? — ek velocity (baad mein momentum se swap hogi).

Neeche sab kuch machinery hai in do facts ko precisely likhne aur ek description ko doosri mein convert karne ke liye.

2. Generalized coordinate $q_i$

Picture: figure s01 ke pendulum ke liye, ek number — straight-down se angle $\theta$ — bob ki location completely fix karta hai. Toh $\theta$ ek generalized coordinate hai. Humein $x$ aur $y$ alag-alag nahi chahiye; wire/rod pehle se hi unhe constrain karta hai.

• Letter $q$ traditional symbol hai.
• Chhota subscript $i$ sirf ek label hai — ek name tag. Agar ek system ko describe karne ke liye teen numbers chahiye, hum unhe $q_1,q_2,q_3$ kehte hain. Symbol $q_i$ ka matlab hai "woh $i$-waan wala, jo bhi tumhe pasand ho".

3. Dot: ${\dot{q}}_i$ ka matlab rate of change

Dot kahan se aata hai — derivative. Maano coordinate time $t$ pe $q$ hai aur thodi si der $\Delta t$ baad $q+\Delta q$. Change ki average speed hai $\Delta q/\Delta t$. $\Delta t$ ko zero ki taraf shrink karo aur yeh ratio ek number pe settle ho jaata hai — instantaneous rate. Woh limiting number wahi hai jo dot denote karta hai: $\dot{q}={\lim }_{\Delta t\to 0}\frac{\Delta q}{\Delta t}=\frac{dq}{dt}.$

Yeh tool kyun aur koi nahi? Humein jaanna hai kitni tezi se coordinate abhi move kar raha hai, poore second ke average pe nahi. Sirf limit — derivative — ek honest "abhi" answer deta hai. Wahi precisely woh sawaal hai jiske liye derivative invent ki gayi thi.

Picture: $q$ ko time ke against plot karo. Dot us curve ki steepness (slope) hai. Tezi se upar $\Rightarrow$ bada positive $\dot{q}$; flat $\Rightarrow$ $\dot{q}=0$; neeche dhalta $\Rightarrow$ $\dot{q}$ negative.

4. Partial derivative $\frac{\partial }{\partial {\dot{q}}_i}$

Parent note mein $\partial L/\partial {\dot{q}}_i$ jaisi cheezein likhi hain. Yahan woh symbol, zero se.

Kyun zaroori hai: Lagrangian $L$ ek saath $q$, $\dot{q}$, aur time $t$ pe depend karta hai. Momentum find karne ke liye humein $L$ ka sirf velocity ke liye response isolate karna hoga. Partial derivative woh akela tool hai jo "ek input ka response, baaki freeze" ka jawaab deta hai.

Picture: $L$ ko ek landscape ki height socho jiske floor ke do directions hain $q$ (east) aur $\dot{q}$ (north). Partial $\partial L/\partial \dot{q}$ woh slope hai jo tum due north chalne pe feel karte ho, kisi bhi east–west tilt ko ignore karke.

5. Lagrangian $L$

• $T$ tab bada hota hai jab cheezein tezi se chalti hain: e.g. $T=\frac{1}{2}m{\dot{x}}^2$. Velocity pe depend karta hai.
• $V$ stored energy hai jo kahan ho pe depend karti hai (height, spring stretch): $V(q)$. Usually velocity-free.

$T-V$ kyun, $T+V$ kyun nahi? Woh minus sign wahi hai jo Euler–Lagrange machinery (neeche) ko Newton ke laws reproduce karne deta hai — yahan faith pe lo; yeh Lagrangian Mechanics mein develop kiya gaya hai. Is topic ke liye, $L$ simply woh raw material hai jis se Hamiltonian bana hai.

6. Conjugate momentum $p_i$

Picture (s03 reuse): $p_i$ exactly $L$-landscape ka north-slope hai. $L$ ki velocity pe zyada depend karna $\Rightarrow$ bada momentum.

Ordinary case ke saath sanity check. $L=\frac{1}{2}m{\dot{x}}^2-V(x)$ ke liye, $\dot{x}$ nudge karne se $p=\partial L/\partial \dot{x}=m\dot{x}$ milta hai — ordinary "mass times velocity". Toh fancy definition familiar wale ko ek special case ke roop mein contain karti hai.

7. Sum symbol ${\sum }_i$

Kyun zaroori hai: ek system mein kai coordinates ho sakte hain. Plus signs ki lambi chain likhne ki bajay, $\sum$ "ek term per coordinate, sab add karo" ko ek compact symbol mein pack karta hai.

8. Energy words: $T$, $V$, $E$

• $T$ — kinetic energy, woh energy jo kisi cheez ke paas hoti hai kyunki woh move kar rahi hai. Tezi ya bhari cheezein ke liye zyada.
• $V$ — potential energy, position se stored energy: ek utha hua weight, ek stretched spring.
• $E=T+V$ — total energy. Ek system mein jahan koi friction/driving nahi, $E$ constant rehti hai jab motion $T$ aur $V$ ke beech energy slosh karta hai.

Parent note ka punchline — "$H$ aksar $E$ ke barabar hota hai" — sirf tab sense karta hai jab tum teen ko seedha rakhte ho. $E$ physical energy hai; $H$ ek constructed quantity hai jo kabhi kabhi, hamesha nahi, iske saath coincide karta hai.

9. Symbols ko kaam pe lagana (Hamiltonian)

Ab $H={\sum }_i{p}_i{\dot{q}}_i-L$ ka har piece defined hai:

Parent note phir velocity description ko momentum description se trade karta hai. Sab kuch downstream — Legendre Transform, Hamilton's Canonical Equations, Phase Space, Poisson Brackets, Conservation Laws & Noether's Theorem — exactly inhi atoms pe built hai.

10. Foundations topic ko kaise feed karte hain

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