Math 2.0

Mechanics of particles and systems Rigid body and multibody dynamics Example

Rattleback

Builds on Rolling without slippingInertia tensor and principal axes

Example

A rattleback is a rigid body with a curved bottom, often close to half an ellipsoid, resting on a horizontal table. Its principal axes of inertia are turned by a small angle with respect to the directions of greatest and least curvature of its bottom. Spun about the vertical in one direction, it keeps spinning. Spun in the other direction, it starts to rock, slows down, and then spins in the opposite direction.

It is a standard example of a nonholonomic system. The usual model lets the body roll without slipping on the table, which is a constraint on velocities, not on positions. In this model, with no dissipation of energy, a linear stability analysis of steady spinning already shows the asymmetry. Steady spin is stable for one sense of rotation and unstable for the other, and the instability turns spin into rocking.

Remarks

The misalignment is necessary. If the principal axes of inertia are aligned with the directions of curvature, a mirror reflection exchanges the two senses of spin, so both behave alike. Real rattlebacks also lose energy through friction, which is why the reversal is eventually followed by the body coming to rest.

Reading path

To understand this page from the ground up, read these 3 pages in order. The foundations come first.

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Inertia tensor and principal axes, concept, no evidence levelInertia tensor and pr…conceptRolling without slipping, concept, no evidence levelRolling without slipp…conceptRattleback, example, no evidence levelRattlebackexample
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Inertia tensor and principal axes concept
  2. Rolling without slipping concept
  3. Rattleback example

Further reading

  1. Web https://en.wikipedia.org/wiki/Rattleback
About this page
TagFGQZTHDG4C
URIm2://FGQZTHDG4C
Kindexample
WritingW0 stub
MSC70E18, 37J60
IllustratesNonholonomic system
UsesRolling without slipping, Inertia tensor and principal axes
Same as
Changed3 November 2026

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