Three-Dimensional Rigid Body

Tensorial dynamics of the rigid body. Covers: Euler Equations for a Rigid Body.

General idea of the topic

In a rotation about a fixed axis a single number is enough, the moment of inertia. When the axis can change direction, angular momentum and angular velocity are in general no longer parallel, and the moment of inertia becomes a tensor. A spinning top or a freely rotating satellite are examples.

The topic writes Euler's equations, which govern rotation in the principal axes of the body, and uses them to study free rotation and the stability of rotation about each principal axis.

What you will truly understand

By the end you will be able to write Euler's equations in the principal-axis frame and recognize that rotation about the axes of largest and smallest moment of inertia is stable, and about the intermediate one it is not.

You will also know why a book tossed in the air with spin about the intermediate axis flips over.

Recommended learning route

1. Euler's Equations: equations of dynamics in the body-fixed frame of axes, free rotation and stability of the principal axes.

Physical criterion you must acquire

The essential criterion is to write everything in the principal axes, where the inertia tensor is diagonal. In free rotation angular momentum and kinetic energy are conserved, and the angular velocity need not stay constant even without external moment.

Connection with real problems

• rotating satellites and spacecraft.
• spinning tops, gyroscopes and balls.
• tools and thrown bodies.
• dynamics of vehicles and aircraft.

Final goal of the topic

The final goal is to analyze the free rotation of a rigid body in three dimensions with Euler's equations and determine the stability of its principal axes.