M087 · Geometry / collision maps

Chaotic Billiards Arena

Can geometry alone create chaos? Yes. Every table uses the same rule at the wall: angle in equals angle out, with no energy lost. In a circle or an ellipse the bounces stay orderly. In a stadium shape, or with a round obstacle in the middle, two almost identical shots soon head in completely different directions. Only the shape of the boundary changed.

Launch two nearby rays into a circle, an ellipse, a stadium or a square with a central disk; every table reflects them by the same elastic law. Compare exact collisions, the caustics of the integrable tables, boundary sections and the growth of a tiny separation to tell regular bouncing from motion that is sensitive to its starting point.

Reviewed collision $\text{Pending}$

Retained collisions $\text{Pending}$

Primary elapsed time $\text{Pending}$

Boundary arclength fraction $\text{Pending}$

Tangential unit-speed component $\text{Pending}$

Collision-index pair distance $\text{Pending}$

Finite-time growth per collision $\text{Pending}$

Valid tangent prefix $\text{Pending}$

Initial local area determinant $\text{Pending}$

Derivative refinement disagreement $\text{Pending}$

Maximum tangential reflection defect $\text{Pending}$

Maximum local speed-squared defect $\text{Pending}$

Circle or ellipse invariant $\text{Pending}$

Maximum invariant drift $\text{Pending}$

One-flight reversal defect $\text{Pending}$

Primary record status $\text{Pending}$

Watch the motion and compare the cases

Can geometry alone create chaos?

Table: circle / ellipse / stadium / Sinai

\mathcal B

2.00\,

Shape: ellipse stretch / straight half-length / disk radius

\eta

0.80\,

Launch boundary arclength fraction

s_0/\mathcal P

0.14\,

Initial tangential velocity component

p_0

0.37\,

Initial position perturbation exponent

\log_{10}\epsilon

-6.00\,

Requested collisions

N

600.00\,

Seeded atlas orbits

N_{\rm atlas}

10.00\,

Reviewed collision or kick

n/N

0.00\,\%

Drag the orange ring in the boundary section to set the launch position and the tangential velocity; both sliders also work from the keyboard. Playback steps through collision numbers, not a shared physical clock, because the nearby trajectory has its own flight times. Each grey atlas path keeps at most 500 collisions, and the apparatus shows the last 40 flights. The growth rate uses boundary coordinates unwrapped near each point and a tangent vector rescaled to unit length after every collision. If the nearby or tangent record fails, it stops on its own, and the main path may still be valid. The drift readouts summarize the whole retained record.

The ball is an ideal point moving at unit speed. It reflects like a mirror, without friction, from fixed walls given by exact formulas. The Sinai table is a square with one circular scatterer, and the ball can fly long distances along its open corridors. A hit exactly on a corner or a near-grazing hit has no reliable smooth continuation, so the affected record stops there. Growth measured over a finite run is a diagnostic; it does not prove that the motion is ergodic.

Start with the essentials

Focus question: Can geometry alone create chaos?
One-sentence intuition: A conserved quantity, the growth of a small separation and a picture built from finitely many bounces each answer a different question.

Core mathematical model

Same reflection law, different dynamics

\mathbf v_{+}=\mathbf v_{-}-2(\mathbf v_{-}\cdot\mathbf n)\mathbf n,\qquad |\mathbf v_{+}|=|\mathbf v_{-}|

Each straight flight ends where the path first meets a valid wall, found analytically. Flipping the normal does not change the reflection. Corner hits and grazing hits are singular.

Collision coordinates that preserve area

q=\frac{s}{\mathcal P},\qquad p=\mathbf v\cdot\mathbf t,\qquad T^{*}(dq\wedge dp)=dq\wedge dp

The boundary runs with the inside of the table on its left: counterclockwise along the outer wall and clockwise around the obstacle. Arclength runs along the joined walls and is divided by the total perimeter. For the ellipse, arclength comes from high-order numerical integration; the polar angle is not used as a canonical coordinate.

The circle keeps a fixed caustic

L=xv_y-yv_x,\qquad r_{\rm caustic}=|L|

In the unit circle, angular momentum stays constant, so every chord touches the same smaller concentric circle, the caustic. A triangular orbit repeats. When the rotation per bounce is an irrational fraction of a turn, the points fill a line in the section. Dense chords and a busy picture do not mean chaos.

Confocal ellipse and hyperbola

\lambda=b^2v_x^2+a^2v_y^2-(xv_y-yv_x)^2,\qquad \frac{x^2}{a^2-\lambda}+\frac{y^2}{b^2-\lambda}=1

The minor semiaxis is one. The value of the invariant tells whether the caustic is an ellipse or a hyperbola; the separatrix through the foci is the borderline case between them. The caustic is predicted analytically, and the fourth plot checks that the invariant stays fixed along the computed collisions.

A nearby ray and a tangent vector measure different things

\lambda_N=\frac1N\sum_{j=0}^{N-1}\log\|DT(z_j)\widehat{\delta z}_j\|,\qquad \widehat{\delta z}_{j+1}=\frac{DT(z_j)\widehat{\delta z}_j}{\|DT(z_j)\widehat{\delta z}_j\|}

The tangent vector starts along the boundary-position direction and is rescaled to unit length after every bounce. The nearby ray is a separate trajectory a finite distance away; it is never reset, so its separation can saturate. Growth is counted per collision, not per unit of physical time. Distance combines the shortest gap in boundary position with the difference in tangential velocity.

Checking the smooth local map numerically

\det DT\approx1,\qquad D_hT(z)=\frac{T(z+h\mathbf e_i)-T(z-h\mathbf e_i)}{2h}

The determinant shown uses central differences in coordinates unwrapped near the point. How much the result changes when the step is halved is numerical evidence, not a rigorous error bound. Derivatives are not taken across a switch to a different wall or across a seam where the curvature jumps. A positive growth rate over a finite run does not prove the whole table is ergodic.

Common difficulties

Complicated is not chaotic

Typical misconceptionA path that fills the circle densely must be chaotic.

Better mental modelIts caustic stays fixed and its section points lie on a line, so the motion is regular.

Choose the right collision coordinates

Typical misconceptionAny evenly spaced angle around the boundary preserves area in the section.

Better mental modelUse true arclength and the velocity component along the wall at unit speed. On an ellipse, the polar angle does not measure arclength.

Collision count is not time

Typical misconceptionThe two paths are compared at the same moment in time.

Better mental modelThey are paired by collision number, and each path has its own flight times.

A corner has no defined bounce

Typical misconceptionAt a corner, the simulation should just pick some bounce.

Better mental modelAt a corner the wall normal is ambiguous, so the run stops; change the starting state instead. The tangent check can stop earlier than the main trajectory.

Run the experiment

01
Start with a triangle
Choose Periodic triangle and press Review full record.
What to observe: The same three impact points repeat; the circular caustic stays fixed.
02
Switch between confocal caustics
Compare the Elliptic caustic and Hyperbolic caustic presets.
What to observe: Both kinds of caustic belong to integrable motion, and the invariant drifts only slightly.
03
Change only the shape
Compare Stadium sensitivity with Parallel-wall bouncing.
What to observe: The same stadium has a few exceptional regular orbits, while typical orbits are sensitive to where they start.
04
Find where the diagnostics stop working
Try Sinai scatterer, change the perturbation, press Double record length, then choose Undefined corner launch.
What to observe: The separation of the nearby ray saturates, the tangent vector can keep growing, and the corner launch stays stopped because its collision is undefined.