# 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:

${\displaystyle \mathbf {I} {\dot {\boldsymbol {\omega }}}+{\boldsymbol {\omega }}\times \left(\mathbf {I} {\boldsymbol {\omega }}\right)=\mathbf {M} .}$

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:

{\displaystyle {\begin{aligned}I_{1}{\dot {\omega }}_{1}+(I_{3}-I_{2})\omega _{2}\omega _{3}&=M_{1}\\I_{2}{\dot {\omega }}_{2}+(I_{1}-I_{3})\omega _{3}\omega _{1}&=M_{2}\\I_{3}{\dot {\omega }}_{3}+(I_{2}-I_{1})\omega _{1}\omega _{2}&=M_{3}\end{aligned}}}

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.