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 angular velocity ω whose axes are fixed to the body. Their general vector form is

where M is the applied torques and I is the inertia matrix. The vector is the angular acceleration.

In orthogonal principal axes of inertia coordinates the equations 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.

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