Phase space — trajectories, phase portraits

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WHAT is phase space?

WHY two axes? Because the equation of motion is second order, the future is fixed only when you know two numbers. Knowing $x$ alone is not enough — a stretched spring released from rest behaves differently from one whipped past the same point at speed.

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HOW the flow is generated — Hamilton's equations

We derive the rule that moves a point through phase space from first principles.

Start from energy. For a conservative 1-D system, $H(q,p) = \frac{p^2}{2m} + V(q).$

The velocity of the state point has two components, $\dot q$ and $\dot p$. We need each.

Component 1 — $\dot q$. Momentum is defined by $p = m\dot q$, so $\dot q = \frac{p}{m} = \frac{\partial H}{\partial p}.$ Why this step? $\partial H/\partial p = \partial/\partial p\,(p^2/2m) = p/m$, exactly $\dot q$.

Component 2 — $\dot p$. Newton's law says force $= -dV/dq$ equals $\dot p$: $\dot p = -\frac{dV}{dq} = -\frac{\partial H}{\partial q}.$ Why this step? $\partial H/\partial q = dV/dq$ since the kinetic term has no $q$.

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Energy conservation ⇒ trajectory shape

For a time-independent $H$,

= \frac{\partial H}{\partial q}\frac{\partial H}{\partial p} + \frac{\partial H}{\partial p}\left(-\frac{\partial H}{\partial q}\right) = 0.$$ *Why this step?* Substitute Hamilton's equations; the two terms are equal and opposite. > [!formula] Trajectories = energy contours > Each conservative trajectory lies on a curve $H(q,p) = E = \text{const}$. **The phase portrait of a conservative system is just the contour map of $H$.** ![[2.1.13-Phase-space-—-trajectories,-phase-portraits.png]] --- ## WHY a harmonic oscillator gives ellipses (full derivation) $H = \dfrac{p^2}{2m} + \dfrac{1}{2}m\omega^2 q^2$. Set $H=E$: $$\frac{p^2}{2m} + \frac{1}{2}m\omega^2 q^2 = E.$$ Divide by $E$: $$\frac{q^2}{\,2E/(m\omega^2)\,} + \frac{p^2}{\,2mE\,} = 1.$$ *Why this step?* This is the standard ellipse form $\frac{q^2}{a^2}+\frac{p^2}{b^2}=1$ with $$a=\sqrt{\tfrac{2E}{m\omega^2}}\ (\text{max displacement}),\qquad b=\sqrt{2mE}\ (\text{max momentum}).$$ So each energy gives an **ellipse**; bigger $E$ → bigger ellipse. The motion runs **clockwise**: when $q>0$, $\dot p = -m\omega^2 q < 0$ (momentum decreasing). --- ## Fixed points & their character > [!definition] Fixed (equilibrium) point > A point where $\dot q = \dot p = 0$: the state never moves. Occurs where $\partial H/\partial q=0$ and $p=0$ — i.e. at extrema of $V(q)$. - **Minimum of $V$ → center:** nearby trajectories are closed loops (stable, oscillatory). - **Maximum of $V$ → saddle:** trajectories approach along one direction, fly off along another (unstable). The special curve through a saddle separating different motions is the ==separatrix==. --- ## Worked Example 1 — Reading a pendulum portrait A pendulum has $V(\theta) = -mgl\cos\theta$. Classify its phase portrait. 1. **Fixed points:** $dV/d\theta = mgl\sin\theta = 0 \Rightarrow \theta = 0,\pi$. *Why?* Equilibria sit at extrema of $V$. 2. $\theta=0$ is a min of $V$ → **center**: small swings are closed loops. 3. $\theta=\pi$ (bob straight up) is a max → **saddle**: unstable. 4. **Separatrix:** the curve with just enough energy to reach $\theta=\pi$ with zero speed. Inside it → oscillation (libration, closed loops). Outside → the bob whirls over the top (rotation, wavy open lines). *Why this matters:* the single separatrix cleanly divides "swinging" from "spinning" motion. ## Worked Example 2 — Damped oscillator spirals $\ddot x + 2\gamma\dot x + \omega_0^2 x = 0$. Energy isn't conserved, so trajectories are **not** closed. 1. Compute $\dfrac{dE}{dt}$ for $E=\tfrac12 \dot x^2 + \tfrac12\omega_0^2 x^2$: $$\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.$$ *Why this step?* Substitute $\ddot x$ from the equation; the $\omega_0^2$ terms cancel. 2. Energy steadily **decreases**, so the trajectory crosses ever-smaller ellipses → an **inward spiral** ending at the origin (a stable focus/sink). ## Worked Example 3 — Sketching from $V(q)$ alone (Feynman test) Given $V(q) = q^4 - q^2$ (double well). 1. $V'=4q^3-2q=0 \Rightarrow q=0,\ \pm\tfrac{1}{\sqrt2}$. 2. $q=\pm 1/\sqrt2$ are minima → **centers** (loops in two wells). 3. $q=0$ is a max → **saddle**. The figure-eight separatrix encircles each well; high-energy orbits enclose both. --- > [!mistake] Steel-manned errors > **"Phase axes are $x$ vs $t$."** *Why it feels right:* we usually plot motion against time. *The fix:* phase space plots $p$ (or $\dot x$) vs $x$ — **time is hidden** as the parameter along the curve. A point on a phase loop tells you state, not the clock reading. > > **"Trajectories can cross at a normal point."** *Feels right* because graphs of functions cross all the time. *Fix:* determinism forbids it — one state, one future. Crossings happen only at fixed points (zero velocity) or in projections of higher-D systems. > > **"Bigger loop = faster everywhere."** *Feels right:* bigger energy. *Fix:* speed in phase space is $|\dot q,\dot p|$ and varies *around* the loop — the point sweeps fast where momentum is large, slow near turning points. --- > [!recall]- Feynman: explain to a 12-year-old > Imagine a map where left–right tells you **where** a swing is, and up–down tells you **how fast** it's moving. The swing's life becomes a dot on this map. As it swings back and forth, the dot goes round and round a loop — never stopping, never crossing its own path. A gentle swing makes a small loop; a wild one makes a big loop. If you add friction, the loop slowly winds inward like water spiraling down a drain until the dot rests in the middle. So this magic map turns "motion over time" into a single picture you can read instantly. > [!mnemonic] > **"PoP": Position over Position-rate."** And for the flow: **"q follows p-slope, p falls down q-slope"** — $\dot q=+\partial_p H$, $\dot p=-\partial_q H$ (the minus lives with $p$). --- ## #flashcards/physics What two quantities define a state point in 1-D phase space? ::: Position $q$ and momentum $p$ (equivalently $x$ and $\dot x$). Why must phase space have two axes for a 1-D system? ::: The equation of motion is second order, so both position and velocity are needed to determine the future. State Hamilton's equations. ::: $\dot q = \partial H/\partial p$ and $\dot p = -\partial H/\partial q$. Why can't two phase trajectories cross at a regular point? ::: The flow vector is single-valued; a crossing would give one state two different futures, violating determinism. What shape is a harmonic-oscillator trajectory, and why? ::: An ellipse, because $\tfrac{p^2}{2m}+\tfrac12 m\omega^2 q^2 = E$ is an ellipse equation; larger $E$ → larger ellipse. For a conservative system, what curve does a trajectory follow? ::: A contour of constant energy, $H(q,p)=E$, since $dH/dt=0$. What is a center vs a saddle in a phase portrait? ::: Center = minimum of $V$, surrounded by closed loops (stable); saddle = maximum of $V$, with trajectories approaching then diverging (unstable). What is a separatrix? ::: The special trajectory through a saddle that separates qualitatively different motions (e.g. swinging vs whirling pendulum). Why does a damped oscillator spiral inward? ::: $dE/dt=-2\gamma\dot x^2\le 0$; energy decreases, so the orbit crosses ever-smaller energy contours toward a stable focus. Where is time in a phase portrait? ::: Hidden as the parameter along each curve; axes carry no time information directly. --- ## Connections - [[Hamiltonian Mechanics]] — phase space is its native arena. - [[Liouville's Theorem]] — phase-space *volume* is conserved by the flow. - [[Harmonic Oscillator]] — the canonical elliptical portrait. - [[Stability and Fixed Points]] — centers, saddles, foci classification. - [[Pendulum]] — libration/rotation separated by a separatrix. - [[Energy Conservation]] — why trajectories are energy contours. ## 🖼️ 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 ``` ## 🔊 Hinglish (regional understanding) > [!intuition]- Hinglish mein samjho > Dekho, Newton ka equation second-order hota hai — matlab future predict karne ke liye sirf position kaafi nahi, velocity (ya momentum) bhi chahiye. Isliye system ki asli "state" hoti hai $(q, p)$ ki jodi. Phase space ek aisa abstract plane hai jiske axes yehi position aur momentum hain. Ek single point poora present batata hai, aur jo curve woh trace karta hai woh poora past aur future hai. Time chhup jaata hai curve ke andar — woh sirf parameter hai. > > Conservative system mein ek mast property hai: energy constant rehti hai, $H(q,p)=E$. Toh trajectory bas energy ka contour ban jaati hai. Harmonic oscillator ka contour ellipse hota hai — zyada energy, bada ellipse. Pendulum mein centre (stable, neeche wala point) aur saddle (upar wala point, unstable) hote hain, aur ek special curve hoti hai jise separatrix bolte hain jo "jhoolna" (libration) aur "ghoomna" (rotation) ko alag karti hai. Damping aaye toh energy girti hai, $dE/dt = -2\gamma\dot x^2 \le 0$, isliye trajectory andar ki taraf spiral karke origin pe ruk jaati hai. > > Sabse important rule yaad rakho: do trajectories kabhi normal point pe cross nahi karti — kyunki har point pe flow ka direction fixed hai, aur agar cross karti toh ek state ke do future ho jaate, jo determinism ke against hai. Isliye phase portrait dekhte hi pura physics samajh aa jaata hai — bina equation solve kiye geometry se hi motion ka nature pata chal jaata hai. Yahi 80/20 hai: $V(q)$ ka graph dekho, minima = centers, maxima = saddles, aur portrait sketch ho gaya. ![[audio/2.1.13-Phase-space-—-trajectories,-phase-portraits.mp3]]