Phase space — trajectories, phase portraits: Visual walkthrough

2.1.13 · D2 · Physics › Analytical Mechanics › Phase space — trajectories, phase portraits

Step 1 — Do numbers, ek nahin, "abhi" ko describe karte hain

KYA. Ek block lo jo mass $m$ ka hai aur ek spring par hai. Ek pal mein hum do cheezein likhte hain: yeh kahan hai, aur kitni tezi se move kar raha hai.

KYUN. Newton ka law $m \overset{x}{¨} = F$ second order hai — ismein $\overset{x}{¨}$ hai, ek acceleration. Aisi equation ka future predict karne ke liye tumhe do starting numbers feed karne padte hain: ek position aur ek velocity. Ek kaafi nahin hai. Ek spring jo $x = 2$ par rest se release hoti hai woh alag move karti hai us spring se jo $x = 2$ se tezi se guzar rahi ho. Toh "abhi" ka honest description ek pair hai.

Hum position $x$ aur momentum $p = m \overset{x}{˙}$ use karte hain. Yahan:

$x$ = rest point se displacement (metres), $+$ (stretched) ya $-$ (compressed) ho sakta hai.
$\overset{x}{˙}$ = velocity (kitni tezi se $x$ change hota hai), dot ka matlab hai "time mein change ki rate".
$p = m \overset{x}{˙}$ = mass times velocity — ek rescaled velocity jo baad ki algebra ko clean banata hai.

PICTURE. Neeche, block apni spring par baayi taraf baitha hai. Daahini taraf hum ek fresh blank plane kholte hain: horizontal axis $x$ hai, vertical axis $p$ hai. Block ka poora "abhi" us plane par ek dot mein simat jaata hai.

Figure — Phase space — trajectories, phase portraits

Step 2 — Dot ko ek velocity do: flow arrows

KYA. Dot frozen nahin hai. Jaise time tikta hai, $x$ aur $p$ dono change hote hain, toh dot drift karta hai. Hum usse ek arrow lagate hain jo dikhata hai yeh kis direction mein move karta hai aur kitni tezi se.

KYUN. Ek point ko move karne ke liye uski velocity chahiye. Dot $(x, p)$ plane mein rehta hai, toh uski velocity ke do parts hain: $x$ kitni tezi se change hota hai ( $\overset{x}{˙}$ ) aur $p$ kitni tezi se change hota hai ( $\overset{p}{˙}$ ). Hum dono ko physics se compute karte hain.

$\overset{x}{˙} = \frac{p}{m}$ . Term by term padho: $\overset{x}{˙}$ (position ki rate) equals $p$ (momentum) divided by $m$ (mass). Yeh sirf $p = m \overset{x}{˙}$ ko ulta karna hai. Yeh kehta hai: zyada momentum ⇒ dot ka tezi se sideways drift hona.
$\overset{p}{˙} = F = - k x$ . Yahan $\overset{p}{˙}$ (momentum ki rate) spring force $- k x$ ke barabar hai. $k$ spring ki stiffness hai; minus kehta hai force hamesha $x = 0$ ki taraf wapas point karti hai. Toh ek stretched spring ( $x > 0$ ) $p$ ko neeche push karti hai; ek compressed spring ( $x < 0$ ) $p$ ko upar push karti hai.

Milaakar har point par arrow yeh chhota vector hai $(\overset{x}{˙}, \overset{p}{˙}) = (\frac{p}{m}, - k x)$ .

PICTURE. Hum yeh arrow kaafi saare points par drop karte hain aur ek poora field of arrows milta hai — phase flow. Notice karo kaise yeh origin ke around curl karte hain: upar tezi se daahine, daahini taraf neeche, neeche tezi se baayein, baayein taraf upar.

Step 3 — Ek arrow se doosre tak follow karo: ek curve banta hai

KYA. Dot ko kahin se shuru karo, maan lo stretched aur rest par: $(x, p) = (a, 0)$ . Use thoda sa apne arrow ke saath step karo, arrow recompute karo, phir step karo. Repeat karo.

KYUN. Ek phase trajectory exactly woh curve hai jo tumhe milti hai jab tum hamesha local arrow ke saath move karte ho. Formally yeh woh curve hai jo har jagah Step 2 ke flow field ke saath tangent ho. Hum ise trace karte hain uski shape dekhne ke liye prove karne se pehle.

Signs track karna ek lap ki kahani batata hai:

$(a, 0)$ par start karo: $\overset{p}{˙} = - k a < 0$ , toh $p$ negative dive karta hai → dot neeche move karta hai.
Ab $p < 0$ : $\overset{x}{˙} = p / m < 0$ , toh $x$ decrease karta hai → dot baayein move karta hai.
$(0, - p_{m a x})$ par pahuncho: fastest point, spring rest length par, saari energy motion hai.
Continue karo: $x < 0$ se $\overset{p}{˙} = - k x > 0$ milta hai, toh $p$ wapas upar climb karta hai → dot upar aur daahine jaata hai.

PICTURE. Dot ek closed loop sweep karta hai, clockwise ghoomta hua. Yeh $(a, 0)$ par wapas aata hai aur hamesha ke liye repeat karta hai — yeh oscillation hai, geometry ke roop mein drawn.

Step 4 — Woh hidden law jo loop band karta hai: energy

KYA. Ab hum prove karte hain ki loop band hota hai — ki dot exactly wapas aata hai, kabhi spiral nahin karta. Tool hai energy.

KYUN ek naya tool? Arrows trace karna ek loop suggest karta hai lekin yeh kabhi prove nahin karta ki dot ek hi curve par rehta hai. Humein ek ऐसी quantity chahiye jo motion ke saath constant rahe; uski constant-value curve phir trajectory hogi. Woh quantity total energy hai.

Energy (the Hamiltonian) define karo: $H = kinetic: motion \frac{p ^{2}}{2 m} + potential: spring stretch \frac{1}{2} k x^{2} .$

$\frac{p ^{2}}{2 m}$ = motion ki energy; momentum ke saath badhti hai, hamesha $\geq 0$ .
$\frac{1}{2} k x^{2}$ = stretched/compressed spring mein stored energy; hamesha $\geq 0$ .

Ab dekho $H$ time mein kaise change karta hai. Hum derivative $\frac{d H}{d t}$ use karte hain kyunki ek derivative exactly woh tool hai jo jawab deta hai "kya yeh quantity change ho rahi hai, aur kitni tezi se?" $\frac{d H}{d t} = \frac{p}{m} \overset{p}{˙} + k x \overset{x}{˙} .$ Term by term: har piece ko differentiate karo aur uski rate ( $\overset{p}{˙}$ , $\overset{x}{˙}$ ) se multiply karo — chain rule. Ab Step 2 ka flow substitute karo ( $\overset{p}{˙} = - k x$ , $\overset{x}{˙} = p / m$ ): $\frac{d H}{d t} = \frac{p}{m} (- k x) + k x (\frac{p}{m}) = - \frac{k x p}{m} + \frac{k x p}{m} = 0.$ Dono terms equal aur opposite hain — yeh exactly cancel ho jaate hain. Toh $H$ kabhi change nahin hota.

PICTURE. Energy ek bowl $H (x, p)$ hai jo plane ke upar baitha hai. Bowl ko fixed height $E$ par slice karo toh ek closed rim milti hai; us rim ki shadow floor par humari trajectory hai. Constant energy = constant height = ek closed contour.

Step 5 — Contour ellipse kyun hai (circle kyun nahin)

KYA. Contour $H = E$ lo aur uski actual shape padho.

KYUN. "Closed curve" kaafi nahin hai — hum precise geometry chahte hain taaki uski width aur height label kar sakein. Hum $H = E$ ko textbook ellipse equation mein massage karte hain.

Shuru karo $\frac{p ^{2}}{2 m} + \frac{1}{2} k x^{2} = E$ se. Har term ko $E$ se divide karo taaki right side $1$ ban jaaye: $\frac{x ^{2}}{2 E / k} + \frac{p ^{2}}{2 m E} = 1.$ Master ellipse form $\frac{x ^{2}}{a ^{2}} + \frac{p ^{2}}{b ^{2}} = 1$ se compare karo:

$a = \frac{2 E}{k}$ — half-width, farthest point $x$ tak pahunchta hai (turning point jahan saari energy spring-stretch hai, $p = 0$ ).
$b = 2 m E$ — half-height, sabse bada momentum (yeh $x = 0$ par milta hai, saari energy motion hai).

Ellipse kyun hai aur circle kyun nahin? Circle ke liye $a = b$ chahiye. Yahan $a$ aur $b$ alag units aur alag physics rakhte hain — ek distance hai jo stiffness $k$ se set hoti hai, doosra momentum hai jo mass $m$ se set hota hai. Yeh sirf ittefaq se equal hote hain. Generally dono axes alag hote hain, toh curve ek squashed circle hai: ek ellipse.

PICTURE. Ellipse apne do half-axes mark kiye hue, plus special points: daahine/baayein tips turning points hain ( $p = 0$ , motion momentarily ruk jaati hai); upar/neeche fastest points hain ( $x = 0$ ). Bada $E$ ⇒ dono $a$ aur $b$ badhte hain ⇒ ek bada nested ellipse.

Step 6 — Edge cases: zero par aur extremes par kya hota hai

KYA. Hum corner cases check karte hain taaki koi reader kabhi surprised na ho.

KYUN. Ek derivation tabhi trustworthy hoti hai jab uski boundaries theek behave karein. Hum $E = 0$ , tips, aur energies ke ek stack ko test karte hain.

$E = 0$ (degenerate). Tab $a = b = 0$ : ellipse single point $(0, 0)$ mein simat jaata hai. Physically block spring ki rest length par hamesha ke liye dead still baitha hai. Yeh fixed point hai — wahan flow arrow zero hai. (Dekho Stability and Fixed Points.)
Right tip $x = a, p = 0$ . Momentum zero matlab momentarily move nahin kar raha — lekin wahan arrow seedha neeche point karta hai ( $\overset{p}{˙} = - k a < 0$ ), toh dot reverse karne wala hai. Yeh turning point hai, stopping point nahin.
Top $x = 0, p = b$ . Position zero, momentum maximum: middle se poori speed par guzte hue. Arrow baayein point karta hai ( $\overset{x}{˙} = b / m > 0$ … wait sign: top par $p = b > 0$ toh $\overset{x}{˙} > 0$ , daahine move kar raha hai) — clockwise motion ke saath consistent hai.
Ek saath kai energies. Har $E$ apna nested ellipse deta hai; milaakar yeh phase portrait banate hain, center ke around nested rings ka ek set, jaise ek valley ka contour map.

PICTURE. Middle mein degenerate point, phir badhte $E$ ke liye nested ellipses, tip/top arrows drawn kiye hue jo dikhate hain yeh turning/fastest points hain — center ke alawa kabhi true stops nahin.

Ek-picture summary

Neeche, poori derivation compressed hai: baayi taraf spring, peeche energy bowl, height $E$ par ek horizontal slice, aur uski shadow — ek clockwise ellipse — $(x, p)$ floor par, turning points aur fastest points label kiye hue. Ise left se right padho aur tumne Steps 1–6 replay kar liya.

Recall Feynman retelling — poora walkthrough plain words mein

Ek block on a spring ko describe karne ke liye do numbers chahiye: yeh kahan hai aur kitni tezi se ja raha hai. Un dono ko ek map par point ke roop mein draw karo — left/right position hai, up/down momentum hai. Woh point stuck nahin hai; har jagah ek chhota arrow hai jo kehta hai yeh kis direction mein drift karta hai, aur arrows middle ke around curl karte hain. Arrows follow karo aur point ek loop sweep karta hai, clockwise jaata hua. Loop kyun band hota hai drift hone ki jagah? Kyunki total energy — motion-energy plus spring-energy — kabhi change nahin hoti; aur "energy same rehti hai" ek aisi fence hai jo point cross nahin kar sakta. Woh fence, jab tum algebra likhte ho, ek ellipse nikalta hai: ek taraf spring ki stiffness se set width, doosri taraf mass se set height, toh yeh ek squashed circle hai. Ellipse ke tips woh moments hain jab block freeze karke reverse karta hai; top aur bottom woh moments hain jab yeh full speed se middle se guzarta hai. Zyada energy do aur tumhe same still center ke around ek bada ellipse milta hai. Woh center — total energy zero — woh ek jagah hai jahan block sacchi mein hamesha ke liye rest karta hai.

Recall

Do axes kyun, ek nahin? ::: Equation of motion second order hai, toh tumhe future fix karne ke liye position aur momentum dono chahiye. $\frac{d H}{d t} = 0$ kya banata hai? ::: Hamilton's equations substitute karne se dono terms equal aur opposite ho jaate hain, toh cancel ho jaate hain. Ellipse kyun na ki circle? ::: Half-axes $a = 2 E / k$ aur $b = 2 m E$ alag hain (alag physics/units), toh circle squashed ho jaata hai. $E = 0$ case kya hai? ::: Ellipse origin par single fixed point mein collapse ho jaata hai — block permanent rest par. Dot kis direction mein circulate karta hai aur kyun? ::: Clockwise, kyunki daahini taraf ( $x > 0$ ) restoring force $p$ ko neeche drive karti hai.

Related: Pendulum (same method, closed loops ek separatrix ban jaate hain) · Stability and Fixed Points · Liouville's Theorem.