Foundations — Phase space — trajectories, phase portraits

2.1.13 · D1 · Physics › Analytical Mechanics › Phase space — trajectories, phase portraits

Yeh page assume karta hai ki tumhe abhi kuch nahi pata. Isse pehle ki tum parent topic padh sako, har woh symbol jo wahan milega, pehle yahan earn karna hoga. Neeche, har idea usse pehle wale idea pe build hota hai — upar se neeche padho, kuch skip mat karo.

0. Kahani ke characters

Parent note in symbols use karta hai: $x$, $q$, $t$, $\dot{x}$, $\dot{q}$, $p$, $m$, $\omega$, $V$, $E$, $H$, $\partial$, letter ke upar dot, aur "vector field" ke arrows. Inhe hum ek ek karke define karenge.

1. Position — $x$ (aur iska cousin $q$)

Parent note kabhi kabhi $x$ ki jagah $q$ likhta hai. Yahan dono ka matlab ek hi hai: ek generalized coordinate. Letter $q$ bas ek thoda zyada grown-up naam hai jo tab bhi kaam karta hai jab "position" ek angle ho, ek length ho, ya kuch bhi jo system ko locate kare. Ek swing ke liye, $q$ angle $\theta$ ho sakta hai; ek wire pe bead ke liye, $q=x$.

2. Time aur dot — $t$ aur $\dot{x}$

Toh $\dot{x}$ (padho "$x$-dot") ka matlab hai "$x$ kitni tezi se change ho raha hai" — yeh velocity hai.

• $\dot{x}>0$: bead seedhi taraf move kar raha hai.
• $\dot{x}<0$: bead ulti taraf move kar raha hai.
• $\dot{x}=0$: bead ek pal ke liye rest mein hai (jaise ball apni throw ke top pe).

Doosra dot, $\ddot{x}$, hai "rate ki rate" — velocity khud kitni tezi se change hoti hai. Yeh acceleration hai. Do dots = acceleration.

3. Mass aur momentum — $m$ aur $p$

Topic $\dot{x}$ ki jagah $p$ kyun chahta hai: yeh "position" aur "motion" ke beech ki deep symmetry dikhata hai (tum $\dot{q}$ aur $\dot{p}$ ko beautifully mirrored equations mein baitha hua dekhoge). Lekin agar yeh dara raha hai, toh bas $p$ ko "velocity in disguise" samjho.

4. Do numbers kyun chahiye — second-order idea

Newton ka law kehta hai $\text{force}=m\ddot{x}$. Notice karo do dots: yeh law tumhe acceleration batata hai, directly position nahi.

Woh pair $(x,\dot{x})$, ya equivalently $(q,p)$, state kehlaata hai. Yahi sabse important reason hai ki phase space do axes ke saath kyun draw ki jaati hai.

5. Potential energy $V$ aur total energy $E$

Do facts jo tum baar baar use karoge:

• Valley bottom ($V$ ka minimum) ek resting place hai jis taraf tum wapas girte ho → stable.
• Hilltop ($V$ ka maximum) ek resting place hai jis taraf se tum door girte ho → unstable.

$E=\underset{\text{kinetic}}{\underbrace{\frac{1}{2}m{\dot{x}}^2}}+\underset{\text{potential}}{\underbrace{V(q)}}.$

6. Energy bookkeeper — $H$

$H(q,p)=\frac{p^2}{2m}+V(q).$

Kinetic energy ko $\frac{p^2}{2m}$ kyun likhte hain? Kyunki $p=m\dot{x}$, toh $\dot{x}=p/m$ aur $\frac{1}{2}m{\dot{x}}^2=\frac{1}{2}m(p/m{)}^2=\frac{p^2}{2m}$. Same energy, ab do state variables mein express ki gayi — exactly woh pair jo phase space plot karta hai. Yeh bookkeeping Hamiltonian Mechanics ka dil hai.

7. Curly-$d$ — $\partial$ (partial derivative)

Parent note $\partial H/\partial p$ jaisi cheezein likhta hai. Curly $\partial$ beginners ko dara deta hai, toh yahan zero se samjhate hain.

Yahan $\omega$ (Greek "omega") bas ek fixed number hai jo set karta hai ki spring kitna stiff/fast hai — zyada $\omega$, zyada tezi se wobble. Tum ise Harmonic Oscillator mein poori tarah miloge.

8. Vector field — woh "flow" jiska topic baar baar zikr karta hai

Yeh single picture — arrows ek plane mein bhar rahe hain, patte unke saath drift kar rahe hain — exactly wahi hai jo ek phase portrait hai. Parent note ki poori cheez is image ke upar hai.

Prerequisite map

Ise aise padho: coordinates + time velocity dete hain; velocity + mass momentum dete hain; position aur momentum milkar state banate hain; energy ke pieces Hamiltonian banate hain; partial derivative Hamiltonian ko arrows ke field mein badal deta hai; state-plane pe arrows phase portrait hain. Woh aakhri box woh topic hai — aur yeh Stability and Fixed Points, Liouville's Theorem, aur Pendulum ke darwaze kholta hai.

Equipment checklist

Apne aap ko test karo — tum ready ho jab har reveal obvious lagey.