Euler's equations (rigid body dynamics)

In classical mechanics, Euler's rotation equations are a vectorial quasilinear first-order ordinary differential equation describing the rotation of a rigid body, using a rotating reference frame with its axes fixed to the body and parallel to the body's principal axes of inertia. Their general form is:

where M is the applied torques, I is the inertia matrix, and ω is the angular velocity about the principal axes.

In three-dimensional principal orthogonal coordinates, they become:

where Mk are the components of the applied torques, Ik are the principal moments of inertia and ωk are the components of the angular velocity about the principal axes.