---
title: "Chaotic Billiards Arena Interactive Lab | Sandbox Physics"
description: "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."
canonical_url: "https://sandboxphysics.com/labs/chaotic-billiards/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/chaotic-billiards/"
last_modified: "2026-10-22"
---

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.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

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.

Physics tutorial

## Geometry organizes conservative motion

**Background** In a mathematical billiard, a ball flies straight and bounces elastically off the walls. Recording only where it hits the wall and its velocity along the wall turns the motion into a map from one bounce to the next.

**Why it matters** Changing the shape of the table can change whether trajectories are stable, even though the reflection law stays the same.

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 misconception** A path that fills the circle densely must be chaotic.

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

#### Choose the right collision coordinates

**Typical misconception** Any evenly spaced angle around the boundary preserves area in the section.

**Better mental model** Use 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 misconception** The two paths are compared at the same moment in time.

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

#### A corner has no defined bounce

**Typical misconception** At a corner, the simulation should just pick some bounce.

**Better mental model** At 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

1. 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.

2. 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.

3. 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.

4. 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.

Physical conclusion

Use exact invariants, boundary sections and growth measurements together, and keep in mind what each one cannot show.

Motion & dynamics · Starter path

## From a falling ball to an orbit

Start by measuring gravity and splitting motion into components. Then add contact forces and collisions, and follow the same laws all the way to an orbit.

[Start here → **Free Fall & Measuring Gravity** Capture a few drops and fit the camera positions to estimate gravitational acceleration. Then add air damping and watch the fit drift from the truth. Observe · 14 min](https://sandboxphysics.com/labs/free-fall/)

1. [1 Free Fall & Measuring Gravity](https://sandboxphysics.com/labs/free-fall/)
2. [2 Projectile Flight Designer](https://sandboxphysics.com/labs/projectile-flight/)
3. [3 Inclined Plane & Friction](https://sandboxphysics.com/labs/inclined-plane/)
4. [4 One-Dimensional Collision Bench](https://sandboxphysics.com/labs/collision-bench/)
5. [5 Mass–Spring Oscillator](https://sandboxphysics.com/labs/mass-spring/)
6. [6 How Orbits Take Shape](https://sandboxphysics.com/labs/orbit/)

[View the full path →](https://sandboxphysics.com/topics/mechanics/#starter-path)