Hamilton's equations of motion — Hinglish notes

2.1.12 · Physics › Analytical Mechanics

1. Hamiltonian KYA hai? (scratch se derivation)

Hum Lagrangian $L (q_{i}, \overset{q}{˙}_{i}, t)$ se shuru karte hain. Euler–Lagrange equations hain $\frac{d}{d t} \frac{\partial L}{\partial q ˙ _{i}} - \frac{\partial L}{\partial q _{i}} = 0.$

Variables cleanly kaise switch karein? Hum Legendre transform use karte hain. Trick yeh hai: agar koi function kisi variable $\overset{q}{˙}$ par depend karta hai, lekin hum chahte hain ki woh uski slope $p = \partial L / \partial \overset{q}{˙}$ par depend kare, toh hum ek nayi function banate hain jiska differential $d \overset{q}{˙}$ nahi rakhta.

2. Hamilton's equations DERIVE KARNA (asli baat)

$H = \sum_{i} p_{i} \overset{q}{˙}_{i} - L$ ka total differential lo: $d H = \sum_{i} (\overset{q}{˙}_{i} d p_{i} + p_{i} d \overset{q}{˙}_{i}) - \sum_{i} (\frac{\partial L}{\partial q _{i}} d q_{i} + \frac{\partial L}{\partial q ˙ _{i}} d \overset{q}{˙}_{i}) - \frac{\partial L}{\partial t} d t .$

Cancel hone ke baad, aur $\partial L / \partial q_{i} = \overset{p}{˙}_{i}$ use karke (Euler–Lagrange se, kyunki $\overset{p}{˙}_{i} = \frac{d}{d t} \frac{\partial L}{\partial q ˙ _{i}} = \frac{\partial L}{\partial q _{i}}$ ): $d H = \sum_{i} \overset{q}{˙}_{i} d p_{i} - \sum_{i} \overset{p}{˙}_{i} d q_{i} - \frac{\partial L}{\partial t} d t .$

Lekin $H (q, p, t)$ ke differential ki definition se: $d H = \sum_{i} \frac{\partial H}{\partial q _{i}} d q_{i} + \sum_{i} \frac{\partial H}{\partial p _{i}} d p_{i} + \frac{\partial H}{\partial t} d t .$

Independent differentials $d q_{i}, d p_{i}, d t$ ke coefficients match karo:

Minus sign kyun? Yeh purely $- \overset{p}{˙}_{i} d q_{i}$ ko $+ \frac{\partial H}{\partial q _{i}} d q_{i}$ se match karne se aata hai. Yeh asymmetric sign hi phase-space flow ko area-preserving banata hai (Liouville) — yeh ek feature hai, koi typo nahi.

3. $H$ ka Conservation (Pehle Forecast, Phir Verify)

4. Worked Example A — 1D particle ek potential mein

System: $L = \frac{1}{2} m \overset{x}{˙}^{2} - V (x)$ .

Momentum. $p = \frac{\partial L}{\partial x ˙} = m \overset{x}{˙}$ . Yeh step kyun? Yeh woh variable define karta hai jo $\overset{x}{˙}$ ko replace karega, aur hume $\overset{x}{˙} = p / m$ invert karne deta hai.
Hamiltonian. $H = p \overset{x}{˙} - L = p \cdot \frac{p}{m} - (\frac{1}{2} m (\frac{p}{m})^{2} - V) = \frac{p ^{2}}{2 m} + V (x)$ . Yeh step kyun? $\overset{x}{˙} = p / m$ har jagah substitute karna $H$ ko sirf $(x, p)$ mein express karta hai — yahan yeh total energy $T + V$ ke barabar hai.
Equations. $\overset{x}{˙} = \frac{\partial H}{\partial p} = \frac{p}{m}, \overset{p}{˙} = - \frac{\partial H}{\partial x} = - \frac{d V}{d x} .$ Yeh step kyun? Doosra equation bas $m \overset{x}{¨} = - V^{'} (x)$ hai — Newton's second law phir se nikalti hai, consistency confirm hoti hai.

5. Worked Example B — Simple Harmonic Oscillator

$V = \frac{1}{2} m ω^{2} x^{2}$ , toh $H = \frac{p ^{2}}{2 m} + \frac{1}{2} m ω^{2} x^{2}$ . $\overset{x}{˙} = \frac{p}{m}, \overset{p}{˙} = - m ω^{2} x .$ Interesting kyun hai? $p$ eliminate karo: $\overset{x}{¨} = - ω^{2} x$ . Phase space $(x, p)$ mein trajectory constant energy ka ek ellipse hai — orbit constant area $\oint p d x = 2 π E / ω$ enclose karta hai, jo action quantization ka ek hint hai.

6. Worked Example C — Frictionless wire bead par mass / generic check

Abstract recipe dikhao ki Newton mein reduce hoti hai jab $H = T + V$ aur $p = m \overset{q}{˙}$ ho; structure isko guarantee karta hai.

Common mistake Common errors ko steel-man karna

Error 1: " $H$ hamesha total energy $E$ hota hai." Kyun sahi lagta hai: Example A mein yeh literally $T + V$ tha. Fix: $H = E$ tabhi hota hai jab (i) kinetic energy velocities mein quadratic ho AUR (ii) coordinate transformation time-independent ho (scleronomic). Rotating frame ya time-dependent constraints ke liye, $H \neq = E$ . Aur $H$ conserved ho sakta hai phir bhi $E$ ke barabar nahi, ya $E$ ke barabar ho lekin conserved nahi.

Error 2: "Bas $\overset{q}{˙}$ ko $p / m$ se replace karo ek constant rule ki tarah." Kyun sahi lagta hai: Free particles ke liye kaam kiya. Fix: Tumhe specific Lagrangian ke liye $p_{i} = \partial L / \partial \overset{q}{˙}_{i}$ invert karna padega. Cross terms ke saath (e.g. magnetic $L = \frac{1}{2} m \dot{r}^{2} + q A \cdot \dot{r}$ ), $p = m \dot{r} + q A \neq = m \dot{r}$ .

Error 3: $\overset{p}{˙} = - \partial H / \partial q$ mein minus sign bhool jaana. Kyun sahi lagta hai: $\overset{q}{˙} = + \partial H / \partial p$ mein plus hai, toh symmetry "suggest" karti hai ki har jagah plus ho. Fix: Minus Legendre algebra se forced hai aur essential hai — iske bina energy conserved nahi rahti.

Recall Feynman style: ek 12-saal ke bachche ko explain karo

Socho tum ek swing karte ball ko describe karte ho, "woh kahan hai aur kitni tez hai" se nahi, balki "woh kahan hai aur kitna push carry karta hai" se (woh push momentum hai). Hamilton ne do khoobsurat simple rules dhunde: position kaise change hoti hai depend karta hai ki energy push ke saath kaise change hoti hai, aur push kaise change hota hai depend karta hai ki energy position ke saath kaise change hoti hai — ek minus sign ke saath taaki kuch bhi runaway na kare. Position sideways aur push upar plot karo, aur ball forever ek closed loop trace karta hai, jaise ghadi ki sui. Loop ka size kabhi nahi badalta — yahi energy constant rehna hai.

Active-Recall Flashcards

q_{i}

ke conjugate generalized momentum kya hai?

p_{i} = \partial L / \partial \overset{q}{˙}_{i}

Legendre transform ke zariye Hamiltonian define karo.

H = \sum_{i} p_{i} \overset{q}{˙}_{i} - L

, jahan

\overset{q}{˙}_{i}

(q, p)

ke terms mein express kiya gaya ho.

Hamilton ke do canonical equations batao.

\overset{q}{˙}_{i} = \partial H / \partial p_{i}

aur

\overset{p}{˙}_{i} = - \partial H / \partial q_{i}

d H

mein

d \overset{q}{˙}_{i}

kyun gayab ho jaata hai?

Kyunki

\partial L / \partial \overset{q}{˙}_{i} = p_{i}

hai, toh

p_{i} d \overset{q}{˙}_{i}

\frac{\partial L}{\partial q ˙ _{i}} d \overset{q}{˙}_{i}

ko cancel kar deta hai — Legendre transform.

Trajectory ke along

d H / d t

kya hai?

d H / d t = \partial H / \partial t

; isliye

H

conserved hai agar explicit

t

nahi hai.

H

total energy

E

ke barabar kab hota hai?

Jab KE velocities mein quadratic ho AUR constraints time-independent hon (scleronomic).

H = p^{2} /2 m + V (x)

ke liye equations of motion kya hain?

\overset{x}{˙} = p / m

\overset{p}{˙} = - V^{'} (x)

, jo

m \overset{x}{¨} = - V^{'}

ke equivalent hain.

\overset{p}{˙} = - \partial H / \partial q

mein minus sign kyun hai?

Yeh

d H

mein differentials match karne se forced hai; yeh phase-space flow ko energy-preserving banata hai.

\partial H / \partial t

aur

\partial L / \partial t

ke beech kya relation hai?

\partial H / \partial t = - \partial L / \partial t

n

second-order Lagrange equations ki jagah kitne first-order equations aate hain?

Phase space mein

2 n

first-order equations.

Connections

Lagrangian Mechanics — woh starting point jahan se hum transform karte hain.
Legendre Transform — woh mathematical engine jo $\overset{q}{˙} \leftrightarrow p$ swap karta hai.
Phase Space and Liouville's Theorem — jahan $2 n$ first-order flow rehta hai.
Poisson Brackets — compact rewriting $\dot{f} = {f, H} + \partial_{t} f$ .
Canonical Transformations — symmetries jo Hamilton's form preserve karti hain.
Hamilton-Jacobi Theory — agla abstraction, optics aur QM ka bridge.
Noether's Theorem — cyclic coordinate $\Rightarrow$ conserved $p_{i}$ .