Euler's equations (rigid body dynamics)

The Euler's equations or Euler's equations are centrifugal motion equations for the rotation of a rigid body. They are differential equations in the main -axis system with the angular velocity as a variable and the principal moments of inertia as coefficients.

The Euler's equations are not to be confused with the Eulerian angles describing the orientation of a body-fixed coordinate system with respect to a space-fixed coordinate system.

Derivation

The Euler's equations follow from the equation of motion of the angular momentum, which is given by

Wherein the angular momentum, the inertia tensor, and the sum of the action of external torque to the body in inertial space is fixed.

By transforming into the principal axis system of the inertial system in general time-dependent inertia tensor is time independent and takes the form

Of. transformed to this

The angular momentum gets by the very simple form

The angular momentum is determined by the transformation of the reference system to

Here, unlike in the first line, to be understood as the time derivative of the angular momentum in the body-fixed coordinate system.

Componentwise formulated this forms the Eulerian equations

In order to exploit the equations of motion, the external torque in the body-fixed system is needed.