Hamiltonian — definition H = Σpᵢq̇ᵢ − L

2.1.11 · Physics › Analytical Mechanics

1. Hamiltonian kya hai?

Sabse important rule, jo beginners bhool jaate hain:

2. Scratch se Hamiltonian derive karna (Legendre transform)

$H$ derive karna aur confirm karna ki yeh $(q,p,t)$ ka clean function hai. $L$ mein ek choti change se start karo:

Yeh step kyun? Yeh sirf total differential hai — yeh batata hai ki $L$ apne har argument ke change par kaise respond karta hai.

Ab definition $p_i=\partial L/\partial {\dot{q}}_i$ aur Euler–Lagrange equation $\frac{d}{dt}\frac{\partial L}{\partial {\dot{q}}_i}=\frac{\partial L}{\partial q_i}$, yani ${\dot{p}}_i=\partial L/\partial q_i$ use karo:

Yeh step kyun? Humne awkward partials ko $p_i$ aur ${\dot{p}}_i$ ke terms mein rewrite kiya, jo woh variables hain jo hum actually chahte hain.

$\sum p_{i}d{\dot{q}}_i$ term mein abhi bhi velocity differential hai jise hum eliminate karna chahte hain. Ise flip karne ke liye product rule use karo:

Yeh step kyun? Yahi Legendre trick ka crux hai: $d(p\dot{q})=pd\dot{q}+\dot{q}dp$, toh hum ek $d\dot{q}$ term ko $dp$ term se swap karte hain.

Substitute karo aur rearrange karo:

Yeh step kyun? $dL$ ko left side par le jaana aur group karna dikhata hai ki woh natural quantity jiska differential sirf $dp$, $dq$, $dt$ involve karta hai, exactly $H=\sum p_i{\dot{q}}_i-L$ hai. Iske differential mein koi $d\dot{q}$ nahi — proof hai ki $H$ genuinely $(q,p,t)$ ka function hai.

3. $H$ total energy kab hota hai?

Agar $V$ velocities par depend nahi karta, toh $p_i=\partial L/\partial {\dot{q}}_i=\partial T/\partial {\dot{q}}_i$, isliye

therefore

4. Worked examples

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