The Phase Plane — what shape does a trajectory trace out?

The Phase Plane

Teach me

Speed

Potential and phase planeU(x) above, p vs x below — one shared x scale

trace and point

energy line H

T = E − U

turning-point ties

contours of H

separatrix

arrow field — direction only

patch of nearby states

Guided Inquiry

1 · The phase plane

Let’s find our way around. The phase plane is the lower panel of the scene: x across, p up. One point in it — the representative point — fixes the whole state, and H = p²/2m + U(x) is its energy. U(x) is drawn above it on the same x scale. Try it: click anywhere in the phase plane to place a point, then press Play and watch x and p both change while H holds still to three decimals.

2 · Which way round?

Choose the harmonic oscillator and place a point on the positive x axis. Before pressing play, predict which way the point will go round its ellipse.

3 · Inner or outer first?

Still with the oscillator, place one point on the positive x axis close to the origin and another further out on the same axis, and start them together. Which one gets back to the x axis first?

4 · Below the hump

Choose the double well and place a point at rest near the bottom of the left-hand well, so that its energy is below the top of the hump. What does its trace look like, and what does the whole set of contours at that energy look like?

5 · A trace that never closes

Choose the pendulum and raise the energy until the trace no longer closes but runs right across the plane, staying above the θ axis all the way. What is the bead doing?

6 · Lines that seem to cross

Choose the inverted hill and switch on the contours. Two of them are straight lines crossing at the origin. How does this fit with the rule that two distinct trajectories can never cross?

7 · Carry it further

Here’s one to carry further on your own. Switch on Cloud mode and drop 60 points at once, then set Direction of time to Reversed. What do you think — why can no two points of the patch ever arrive at the same place?

Listen

Controls

SystemHarmonic oscillator

Live state

x—

p—

H—

Clock

One circuit—

Patch

Patch area—

60 points spaced round the rim of a disc of radius 0.08 in (x, p) — the boundary of the patch.

Hamilton's equations: the flow in the phase plane has one direction at every point.

H is constant along the flow, so every trajectory lies on a contour of H.

The period between adjacent turning points, as an integral over the path.

Where both partial derivatives of H vanish, the point is at rest.

Fundamental bracket and Liouville's theorem: the patch stretches and bends, its area does not change.

The five potentials on offer, with m = 1 kg throughout: ω = 1 rad/s, a = 1 m, R = 1 m, g = 9.80665 m/s², λ = 1 s⁻¹.

The Physics

The phase plane · what this sim shows, the physics behind it, and every control

At a glance

Each point (x, p) in the phase plane is a complete state of a particle moving in one dimension: where it is, and how much momentum it has. This sim drops points, or a patch of them, into five potentials and follows them, the picture behind oscillators, pendulums and statistical mechanics.

What you'll learn

How one point (x, p) describes a whole state, and how H = p²/2m + U(x) links the panels.
How the shape of U(x) sets the pattern of motion in the phase plane.
How the time for one circuit depends on the system and on the energy.
How a patch of nearby states moves, and what that does to its area.

The sim's two panels share one x scale: U(x) above, the phase plane below.

What's on screen

Violet curve: the potential U(x), in the upper panel.
Green line: the energy H of the point. The gap down to the violet curve is p²/2m.
Cyan point and trace: the state in the phase plane (lower panel), and the path it has covered.
Grey curves: curves of constant energy (Contours of H). Amber: the separatrix.
Live state: x, p and H of the newest point. Clock and Patch follow.

See itDrop one point on the Oscillator and press Play. Watch for: x and p changing in the Live state as the dot moves.

The controls

Press ⌖ on a row to see where that control sits in the sim. Some controls appear only as you work through the Guided Inquiry.

Control	What it changes	Range
Click in the phase plane	Places a point at that (x, p). The newest four points are kept. In Cloud mode one click drops a ring of 60 points.	lower panel
System	The potential U: Oscillator, Free particle, Double well, Pendulum (a bead on a vertical hoop, with θ in radians and p_θ) or Inverted hill.	5 choices opens on Oscillator
Height of the hump	The height h of the barrier between the two wells. It appears only on the Double well.	0.20–3.00 J opens at 1.00 J
Contours of H	Draws every curve of constant energy behind the point.	on / off
Arrow field	Draws an arrow at each point of the plane showing which way the flow runs. Arrows show direction only, not speed.	on / off
Cloud mode	Each click drops a ring of 60 points on a circle of radius 0.08 instead of one point.	on / off
Direction of time	Runs the same flow forward, or backwards.	Forward / Reversed
Live state	x, p and H of the newest point.	panel
Clock	The time for one circuit of each point's contour, or why it has none.	panel
Patch	The area enclosed by the ring of 60 points, in J s.	panel
▶ Play / ⏸ Pause	Moves the points along the flow, or freezes them.	top bar
Speed	Plays the motion faster or slower on screen. The physics and the periods stay the same.	0.25×–4× opens at 1×
↻ Reset	Puts each point back where you placed it, sets the hump to 1.00 J and pauses. It keeps the system and the toggles.	top bar

Fixed: m = 1 kg, ω = 1 rad/s, a = 1 m, R = 1 m, g = 9.80665 m/s², λ = 1 s⁻¹, and no friction.

A point is a whole state

Once U(x) is fixed, the pair (x, p) settles everything that happens next. Its energy is H = p²/2m + U(x).

The green energy line sits at height H. The gap down to U(x) is p²/2m, so the upper panel shows the momentum of the point below it.

Both x and p change as the point moves, but H does not. Each point stays on its own contour of H.
With m = 1 kg, a point at x = 0.6 m with p = 0.8 kg m/s on the Oscillator has H = ½(0.6)² + ½(0.8)² = 0.500 J.

See itThe same point, x = 0.6 m and p = 0.8 kg m/s, now with the contours on. Watch for: the point staying on one grey contour while H holds at 0.500 J.

The flow has one direction at every point

Hamilton's equations give the velocity of the point in the phase plane at every (x, p):

The drift in x is p/m. The push in p is the slope of the well, with a minus sign.

Above the x axis (p > 0) the point moves right; below it (p < 0) it moves left. A closed contour is therefore run clockwise.
At a turning point p = 0, so ẋ = 0, but ṗ = −dU/dx is generally not zero. The push turns the point round.
Putting the two equations into dH/dt gives zero, so the arrow at each point lies along the contour of H through it.
Each point has exactly one arrow, so two different orbits never cross.

Why H is conserved.

A closed contour, run clockwise.

See itSwitch on the arrow field and the contours, then release a point. Watch for: the point circling clockwise, right across the top half and left across the bottom half.

How long one circuit takes

One circuit is a crossing of the well and the way back. The integral adds up the time spent at each x, where the speed is √(2(E − U)/m).

The period T, and its value for the parabolic well.

Oscillator. The integral gives T = 2π/ω = 6.2832 s on every ellipse, because ω = 1 rad/s. That is special to the parabola.
Other wells. T depends on the energy. In the Double well, a small orbit in one well runs at ω = √(8h/m), and larger orbits take longer.

This slider moves the sim's Height of the hump too.

See itTwo Oscillator points, one near the centre and one far out. Watch for: both Clock readings equal, 6.2832 s.

See itTwo Double-well points in the same well, one with more energy. Watch for: two different Clock readings, the more energetic orbit slower.

Energy decides the kind of motion

Since p²/2m = E − U cannot be negative, a point stays where U(x) ≤ E, and turns round where U(x) = E.

The Double well at the default hump: two centres, one saddle, and the separatrix.

Double well. Below E = h the contour is two separate closed curves, and a point stays on its own. Above h it is one curve round both wells.
Pendulum. Below 2mgR = 19.613 J the bead swings. Above it, p_θ never reaches zero: the bead goes round the hoop and θ wraps from π to −π.
Separatrix. The contour at E = h (or 2mgR) divides the two kinds of motion. Its period is infinite, since the point slows to a crawl near the top. The sim's finite time step cannot hold a point exactly on it, so after a few seconds the point slips off to one side.
Points of rest. Where ∂H/∂x = ∂H/∂p = 0 the state never changes: a centre at a well bottom, a saddle at a hilltop.

On the Inverted hill, a point on the incoming half of the line H = 0 gets ten times closer to the origin every ln10/λ = 2.3026 s, and never arrives.

See itLaunch a Double-well point with E exactly h. Watch for: the point slowing to a crawl as it nears the origin, and the Clock reading "never arrives".

See itGive a Pendulum point p_θ = 7 kg m²/s at θ = 0. Watch for: the trace running across the plane and wrapping at the edge, never turning round.

See itStart an Inverted-hill point on the line H = 0. Watch for: x falling roughly tenfold every 2.3 s.

Area in the plane is conserved

Liouville's theorem: the flow can stretch and bend a patch of states, but it never changes the patch's area.

The Poisson bracket {x, p} = 1, and the area of a patch.

Cloud mode drops 60 points on a circle of radius 0.08 in the (x, p) plane. Its area is π(0.08)² = 0.0201 J s, shown as Patch area.
On the Oscillator every point has the same period, so the patch simply turns round. In the Double well, points on different contours have different periods, so the patch shears into a long thin filament. The area stays 0.0201 J s.
The flow is reversible. With Direction of time on Reversed, the filament folds back towards the disc it started as.

See itRelease a patch in the left-hand well of the Double well. Watch for: the patch stretching out while Patch area stays at 0.0201 J s.

See itLet a patch stretch out for 4 s, then run time backwards for 4 s. Watch for: the stretched patch folding back towards its starting disc.

Where it matters

Statistical mechanics. Counting the microstates of a gas means measuring volume in phase space. Liouville's theorem says the flow preserves that measure.
Particle beams. The area a bunch fills in (x, p) is its emittance. Focusing magnets can squeeze the beam in x, but they spread its momenta; shrinking the area needs beam cooling, which is not Hamiltonian flow.
Small oscillations. Near any well bottom U ≈ ½U″x², so ω = √(U″/m). It is how a molecular bond is modelled as a spring, with the reduced mass in place of m.

Go deeper: why the flow cannot change an area

The flow velocity is (ẋ, ṗ) = (∂H/∂p, −∂H/∂x). Its divergence is ∂²H/∂x∂p − ∂²H/∂p∂x, which is zero for any smooth H.
A flow with zero divergence neither compresses nor expands area. That holds for every H, not only the five here.

Summary

In one line. Each point (x, p) moves along a contour of H = p²/2m + U(x), and the flow that carries it never changes the area of a patch of states.

Key points

A point (x, p) is a complete state, and its energy H is conserved.
The flow is ẋ = p/m, ṗ = −dU/dx, clockwise on closed contours.
The period depends on the energy, except for the parabola, where it is 2π/ω for every orbit.
The energy decides the kind of motion, and the separatrix divides the kinds.

Common mix-ups

"The phase plane draws the path of the particle in space." The particle moves only along x. The vertical axis is its momentum.
"At a turning point the particle stops." Only p = 0 there; ṗ = −dU/dx is generally not.
"Orbits of different energy can cross." Each point has one arrow, so orbits never cross.

Things to try on your own

On the Pendulum, place points at θ = 0 with increasing p_θ. Find where the trace stops closing, and compare H there with 19.613 J.
On the Double well, set the hump to 3 J and place a point at x = 1 m with a very small p. Compare its Clock with 2π/√(8h) = 1.28 s.
Turn on Cloud mode and compare a patch after 10 s on the Oscillator and on the Double well.

Predict before reveal	Each of the five questions is gated; the clock, the contours and the flow arrow appear only after the prediction they would have answered.
Invariant made visible	H sits still on the readout to three decimals while x and p both move — the symplectic leapfrog keeps it there.
One scene, one alignment	U(x) and the phase plane share a single x-to-pixel map, so the energy line meets U exactly above where the trace crosses p = 0.
Misconception targeted structurally	The crossing lines on the hill are drawn only once found, and the point visibly stalls — equal time for every decade closer.
Operational numbers	Three readouts and a clock. Card 1 asks for H against the moving x and p; the clock, the contours and the patch area are instruments for the cards that reach them, and for free exploration afterwards.
Formal layer optional	Hamilton's equations, the period integral and Liouville's theorem live behind the Physics sheet.