Home / Topics / Rotational Motion
Euler's equations
Advanced
Rigid-body rotation about principal axes.
I₁ω̇₁ = (I₂−I₃)ω₂ω₃The rotation of a rigid body about its principal axes: I₁ω̇₁ = (I₂ − I₃)ω₂ω₃ + τ₁, and cyclic permutations. They explain why a tossed book spins stably about two axes but tumbles about the middle one.
The tennis racket theorem.
Valid in the body frame.
In more detail
Euler's equations describe the rotation of a rigid body about its principal axes, linking the components of torque, angular velocity and moment of inertia. They are used for tumbling motion and spacecraft attitude.
Real-world uses
- Spacecraft attitude control uses Euler's equations to predict tumbling.
- Analysing the rotation of asteroids and satellites.
- Engineers studying the tumbling of a satellite use Euler's equations to predict how its spin changes about each principal axis.
Open the interactive lesson, with simulation and quiz →
Frequently asked questions
What are principal axes?
Axes along which the inertia tensor is diagonal, so angular momentum and angular velocity are parallel.
When is the motion simple?
Rotation about a principal axis of a symmetric body is stable and simple.
Why are Euler's equations hard to solve?
They are coupled, so the change in one component depends on the others, and they usually need numerical methods.
Related topics in Rotational Motion
Rotation of the Earth · Tops and bicycles · Rotating frames · Inertia tensor · Principal axes · Torque on a dipole