Phase space — trajectories, phase portraits

2.1.13 · Physics › Analytical Mechanics

Phase Space KYA hai?

DO axes kyun? Kyunki equation of motion second order hai, future tab hi fix hota hai jab aap do numbers jaante ho. Sirf $x$ jaanna kaafi nahi — ek stretched spring jo rest se release ho woh alag behave karti hai us spring se jo same point pe speed se guzri ho.

Flow kaise generate hota hai — Hamilton's equations

Ab hum woh rule derive karte hain jo ek point ko phase space mein move karta hai — first principles se.

Energy se shuru karte hain. Ek conservative 1-D system ke liye, $H (q, p) = \frac{p ^{2}}{2 m} + V (q) .$

State point ki velocity ke do components hain, $\overset{q}{˙}$ aur $\overset{p}{˙}$ . Hume dono chahiye.

Component 1 — $\overset{q}{˙}$ . Momentum define hota hai $p = m \overset{q}{˙}$ se, isliye $\overset{q}{˙} = \frac{p}{m} = \frac{\partial H}{\partial p} .$ Yeh step kyun? $\partial H / \partial p = \partial / \partial p (p^{2} /2 m) = p / m$ , exactly $\overset{q}{˙}$ .

Component 2 — $\overset{p}{˙}$ . Newton ka law kehta hai force $= - d V / d q$ equals $\overset{p}{˙}$ : $\overset{p}{˙} = - \frac{d V}{d q} = - \frac{\partial H}{\partial q} .$ Yeh step kyun? $\partial H / \partial q = d V / d q$ kyunki kinetic term mein $q$ nahi hai.

Energy conservation ⇒ trajectory ka shape

Time-independent $H$ ke liye,

= \frac{\partial H}{\partial q}\frac{\partial H}{\partial p} + \frac{\partial H}{\partial p}\left(-\frac{\partial H}{\partial q}\right) = 0.$$ *Yeh step kyun?* Hamilton's equations substitute karo; do terms equal aur opposite hain. > [!formula] Trajectories = energy contours > Har conservative trajectory ek curve $H(q,p) = E = \text{const}$ par hoti hai. **Conservative system ka phase portrait bas $H$ ka contour map hai.** ![[2.1.13-Phase-space-—-trajectories,-phase-portraits.png]] --- ## Harmonic oscillator ellipses kyun deta hai — poori derivation $H = \dfrac{p^2}{2m} + \dfrac{1}{2}m\omega^2 q^2$. Set karo $H=E$: $$\frac{p^2}{2m} + \frac{1}{2}m\omega^2 q^2 = E.$$ $E$ se divide karo: $$\frac{q^2}{\,2E/(m\omega^2)\,} + \frac{p^2}{\,2mE\,} = 1.$$ *Yeh step kyun?* Yeh standard ellipse form $\frac{q^2}{a^2}+\frac{p^2}{b^2}=1$ hai jisme $$a=\sqrt{\tfrac{2E}{m\omega^2}}\ (\text{max displacement}),\qquad b=\sqrt{2mE}\ (\text{max momentum}).$$ Toh har energy ek **ellipse** deti hai; bada $E$ → bada ellipse. Motion **clockwise** chalti hai: jab $q>0$ ho, $\dot p = -m\omega^2 q < 0$ (momentum ghatta hai). --- ## Fixed points aur unka character > [!definition] Fixed (equilibrium) point > Woh point jahan $\dot q = \dot p = 0$ ho: state kabhi move nahi karti. Yeh wahan hota hai jahan $\partial H/\partial q=0$ aur $p=0$ — yaani $V(q)$ ke extrema par. - **$V$ ka minimum → center:** nearby trajectories closed loops hain (stable, oscillatory). - **$V$ ka maximum → saddle:** trajectories ek direction se approach karti hain, doosri se bhaag jaati hain (unstable). Saddle se guzarne wali special curve jo alag motions ko alag karti hai woh ==separatrix== hai. --- ## Worked Example 1 — Pendulum portrait padhna Ek pendulum ka $V(\theta) = -mgl\cos\theta$ hai. Iske phase portrait ko classify karo. 1. **Fixed points:** $dV/d\theta = mgl\sin\theta = 0 \Rightarrow \theta = 0,\pi$. *Kyun?* Equilibria $V$ ke extrema par hote hain. 2. $\theta=0$ par $V$ minimum hai → **center**: chhote swings closed loops hain. 3. $\theta=\pi$ (bob seedha upar) maximum hai → **saddle**: unstable. 4. **Separatrix:** woh curve jisme exactly itni energy ho ki $\theta=\pi$ zero speed se reach ho sake. Iske andar → oscillation (libration, closed loops). Bahar → bob upar se ghoomta hai (rotation, wavy open lines). *Yeh kyun matter karta hai:* single separatrix "swinging" aur "spinning" motion ko saaf alag karti hai. ## Worked Example 2 — Damped oscillator spirals $\ddot x + 2\gamma\dot x + \omega_0^2 x = 0$. Energy conserved nahi, isliye trajectories **closed** nahi hain. 1. $E=\tfrac12 \dot x^2 + \tfrac12\omega_0^2 x^2$ ke liye $\dfrac{dE}{dt}$ compute karo: $$\dot E = \dot x\ddot x + \omega_0^2 x\dot x = \dot x(-2\gamma\dot x) = -2\gamma\dot x^2 \le 0.$$ *Yeh step kyun?* $\ddot x$ equation se substitute karo; $\omega_0^2$ terms cancel ho jaate hain. 2. Energy steadily **kahmti** hai, isliye trajectory har baar chhoti ellipses cross karti hai → ek **inward spiral** jo origin pe khatam hota hai (ek stable focus/sink). ## Worked Example 3 — Sirf $V(q)$ se sketch karna (Feynman test) Diya gaya $V(q) = q^4 - q^2$ (double well). 1. $V'=4q^3-2q=0 \Rightarrow q=0,\ \pm\tfrac{1}{\sqrt2}$. 2. $q=\pm 1/\sqrt2$ minima hain → **centers** (do wells mein loops). 3. $q=0$ maximum hai → **saddle**. Figure-eight separatrix har well ko encircle karti hai; high-energy orbits dono ko enclose karti hain. --- > [!mistake] Steel-manned errors > **"Phase axes $x$ vs $t$ hain."** *Kyun sahi lagta hai:* hum usually motion ko time ke against plot karte hain. *Fix:* phase space $p$ (ya $\dot x$) ko $x$ ke against plot karta hai — **time hidden** hai curve ke parameter ke roop mein. Phase loop ka ek point state batata hai, clock reading nahi. > > **"Trajectories ek normal point par cross kar sakti hain."** *Sahi lagta hai* kyunki functions ke graphs aksar cross karte hain. *Fix:* determinism ise forbid karta hai — ek state, ek future. Crossings sirf fixed points (zero velocity) par ya higher-D systems ke projections mein hoti hain. > > **"Bada loop = har jagah tez."** *Sahi lagta hai:* zyada energy. *Fix:* phase space mein speed $|\dot q,\dot p|$ hai aur loop ke *around* vary karti hai — point wahan fast sweep karta hai jahan momentum bada ho, turning points ke paas dheema. --- > [!recall]- Feynman: 12-saal ke bachche ko samjhao > Socho ek map jisme left–right batata hai ki swing **kahan** hai, aur up–down batata hai ki woh **kitni tez** ja rahi hai. Swing ki poori zindagi is map par ek dot ban jaati hai. Jab woh aage-peechhe jhoolti hai, dot ek loop mein ghoomta rehta hai — kabhi rukta nahi, kabhi apna path cross nahi karta. Chhota swing → chhota loop; bada swing → bada loop. Agar friction daalo, loop dheere-dheere andar wind hota hai jaise paani drain mein spiral karta hai, tab tak jab tak dot middle mein rest nahi kar leta. Toh yeh magic map "time ke saath motion" ko ek aisi single picture mein badal deta hai jo aap instantly padh sako. > [!mnemonic] > **"PoP": Position over Position-rate.** Aur flow ke liye: **"q follows p-slope, p falls down q-slope"** — $\dot q=+\partial_p H$, $\dot p=-\partial_q H$ (minus $p$ ke saath rehta hai). --- ## #flashcards/physics 1-D phase space mein ek state point kaunsi do quantities define karti hain? ::: Position $q$ aur momentum $p$ (equivalently $x$ aur $\dot x$). 1-D system ke liye phase space mein do axes kyun honi chahiye? ::: Equation of motion second order hai, isliye future determine karne ke liye position aur velocity dono chahiye. Hamilton's equations batao. ::: $\dot q = \partial H/\partial p$ aur $\dot p = -\partial H/\partial q$. Do phase trajectories ek regular point par kyun cross nahi kar sakti? ::: Flow vector single-valued hai; crossing se ek state ke do alag futures ho jaate, jo determinism ka violation hai. Harmonic oscillator trajectory ka shape kya hai, aur kyun? ::: Ellipse, kyunki $\tfrac{p^2}{2m}+\tfrac12 m\omega^2 q^2 = E$ ellipse equation hai; bada $E$ → bada ellipse. Conservative system mein trajectory kaunsi curve follow karti hai? ::: Constant energy ka contour, $H(q,p)=E$, kyunki $dH/dt=0$ hai. Phase portrait mein center aur saddle kya hota hai? ::: Center = $V$ ka minimum, closed loops se ghira (stable); saddle = $V$ ka maximum, trajectories approach karke diverge karti hain (unstable). Separatrix kya hai? ::: Woh special trajectory jo saddle se guzarti hai aur qualitatively alag motions ko alag karti hai (jaise pendulum ka swinging vs whirling). Damped oscillator andar kyun spiral karta hai? ::: $dE/dt=-2\gamma\dot x^2\le 0$; energy kahmti hai, isliye orbit har baar chhote energy contours cross karta hai ek stable focus ki taraf. Phase portrait mein time kahan hai? ::: Har curve ke parameter ke roop mein hidden hai; axes mein directly koi time information nahi hoti. --- ## Connections - [[Hamiltonian Mechanics]] — phase space iska native arena hai. - [[Liouville's Theorem]] — phase-space *volume* flow se conserved rehta hai. - [[Harmonic Oscillator]] — canonical elliptical portrait. - [[Stability and Fixed Points]] — centers, saddles, foci classification. - [[Pendulum]] — libration/rotation jo separatrix se alag hoti hain. - [[Energy Conservation]] — isliye trajectories energy contours hain. ## 🖼️ Concept Map ```mermaid flowchart TD NL[Newton 2nd order law] -->|needs two numbers| STATE[State q,p] STATE -->|all states form| PS[Phase space q,p plane] PS -->|single point| PRESENT[Complete present] PS -->|contains curves| TRAJ[Phase trajectory q of t, p of t] TRAJ -->|many initial conditions| PORTRAIT[Phase portrait / flow] H[Hamiltonian H = p^2/2m + V] -->|partial derivatives| HEQ[Hamilton equations] HEQ -->|q-dot = dH/dp| VF[Velocity vector field] HEQ -->|p-dot = -dH/dq| VF VF -->|tangent curves| TRAJ VF -->|single-valued| NOCROSS[Trajectories never cross] NOCROSS -->|touch only at| FP[Fixed points, vector = 0] H -->|time-independent so dH/dt = 0| ECON[Energy conserved] ECON -->|H = const contours| TRAJ ```