Foundations — Hamilton's equations of motion

2.1.12 · D1 · Physics › Analytical Mechanics › Hamilton's equations of motion

Is page par kuch bhi assume nahi kiya gaya. Agar parent note mein koi symbol use hua tha, toh hum usse yahan pehle ek picture se banate hain. Upar se neeche padho; har block agli cheez ko earn karta hai.

0. Ye saare letters hain kya

Kisi bhi formula se pehle, yeh hai un characters ki list jo tumse milenge. Har ek ko hum neeche theek se define karenge — yeh sirf isliye hai taaki tumhe pata ho kaun aa raha hai.

Ab hum inhe, ek-ek karke, zero se banate hain.

1. Ek number jo move kar sakta hai: variable $q$

Socho ek bead ek wire par slide kar rahi hai, ya ek pendulum jhool raha hai. Yeh kehne ke liye ki woh kahan hai, hume ek number chahiye. Ise $q$ kehte hain.

• Agar bead ek seedhi wire par hai, toh $q$ ek marked zero se uski doori hai — jaise ruler ki reading.
• Agar woh pendulum hai, toh $q$ seedha-neeche se angle ho sakta hai.

Topic ko yeh kyun chahiye? Motion ka har law ultimately yeh baat hai ki $q$ time ke saath kaise badalti hai. $q$ nahi, toh kuch describe karne ko nahi.

Agar kai moving parts hain, toh hum $q_1,q_2,\dots$ likhte hain. Chota number subscript (ek index) kehlaata hai. Hum $q_i$ likhte hain matlab "i-waan wala" — ek placeholder jo sabko ek saath represent karta hai.

2. Change ki rate: dot, $\dot{q}$

Bead ko dekho. Time ke ek tiny slice mein woh thodi si doori move karti hai. Speed hai "doori moved ÷ time taken." Jab hum time slice ko almost zero tak shrink karte hain, yeh ratio ek single number par settle ho jaata hai: instantaneous velocity.

Yeh kaisa dikhta hai? $q$ ko upar aur time $t$ ko sideways plot karo. Bead ki history ek curve hai. Dot $\dot{q}$ us curve ki steepness (slope) hai har instant par — steep matlab fast, flat matlab uss waqt ruk sa gaya.

Do dots, $\ddot{q}$, matlab slope ka slope — velocity kitni tez badalti hai, yaani acceleration.

3. Ek landscape ki slope, ek se zyada direction mein: $\partial$

Ab maan lo ek quantity do cheezon par ek saath depend karti hai — jaise ek pahadi landscape ki height $H$ depend karti hai kitna east ($q$) aur kitna north ($p$) tum khadhe ho: $H(q,p)$.

Agar main puche "pahadi kitni steep hai?", toh mujhe poochna padega "kaun se direction mein steep?"

Yeh kaisa dikhta hai: pahadi par khade ho. Due east ki taraf munh karo aur apne pair ke neeche slope note karo — woh hai $\partial H/\partial q$. Bina hile north ki taraf munh ghumaao — woh nayi slope hai $\partial H/\partial p$. Same jagah, do alag steepnesses.

Topic ko yeh kyun chahiye. Hamilton's equations kehti hain ki system ki velocity $H$ ki ek direction mein slope ke barabar hai, aur momentum ka change minus doosri direction ki slope ke barabar hai. "Chosen direction mein slope" ke bina, tum equations likh bhi nahi sakte.

4. System jo push carry karta hai: momentum $p$

Mass $m$ ke ek plain particle ke liye, momentum hai $p=m\dot{q}$ — bhaari aur tez cheezon mein zyada "push" hoti hai. Lekin Hamilton ko ek zyada general momentum chahiye jo angles, fields, weird coordinates ke liye kaam kare.

Tum abhi $L$ nahi samjhe — woh agli section hai. Abhi yeh thought pakde rakho: momentum $L$ ki velocity direction mein slope hai. Isliye hume pehle $\partial$ chahiye tha.

$p$ ki zarurat kyun hai. Hamilton ki poori trick yeh hai ki velocity $\dot{q}$ use karna band karo aur iska push $p$ use karo. Payoff (parent page par bana) ek bilkul symmetric pair of rules hai $q$ aur $p$ ke liye.

5. Saare parts mein add karna: $\sum$

Agar system mein kai coordinates hain, toh hum often ek term per coordinate add karte hain.

Ek moving part waale system ke liye, ${\sum }_i$ mein ek hi term hai aur tum ise mentally ignore kar sakte ho. Yeh tab kaam aata hai jab kai parts hon.

6. Lagrangian $L$ — jis cheez se hum start karte hain

Parent page $L$ se start hota hai. Yahan woh hai kya, zero se.

Ek difference $T-V$ kyun aur sum kyun nahi? Yeh subtle hai aur tumhe ise "energy" ki tarah memorize nahi karna chahiye. $L$ ek bookkeeping device hai: Nature ek system ko aise move karti hai ki poore path par $L$ ka running total jitna ho sake utna chhota ho (principle of least action, Lagrangian Mechanics se). $T-V$ ka difference exactly woh combination hai jo yeh kaam karta hai. Hum ise yahan apna given starting material maante hain.

Yeh kya depend karta hai. $L$ position, velocity, aur possibly time ka function hai: $L(q,\dot{q},t)$. Woh "aur time" baad mein matter karta hai — agar $L$ secretly $t$ ke saath badalti hai (jaise koi apparatus ko hila raha ho), toh energy conserve nahi ho sakti.

7. Hamiltonian $H$ — jahan hum ja rahe hain

$H$ ko $L$ se Legendre transform naam ki ek swap se banaya jaata hai (iska apna deep-dive hai: Legendre Transform). Hum yahan sirf iska preview dete hain.

Woh picture jo $H$ ko special banati hai: socho $H(q,p)$ ek flat map par ek landscape hai jiska east-axis $q$ hai aur north-axis $p$. Woh map phase space kehlaata hai (Phase Space and Liouville's Theorem). System ki poori life is map par ek point ki wandering hai, aur $H$ ki slopes use batati hain ki aage kahan jaana hai.

8. Saare pieces kaise fit hote hain

Ise ek supply chain ki tarah padho: position aur velocity $L$ ko feed karte hain; velocity direction mein $L$ ka partial derivative momentum $p$ banata hai; $p$, sum, aur $L$ milke $H$ banate hain; aur $H$ ki slopes woh equations of motion hain jiske baare mein poora topic hai.

Equipment checklist

Khud test karo — right side cover karo. Agar koi jawaab surprise kare, toh parent page se pehle woh section dobara padho.