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. Again, note that all quantities are defined in the rotating reference frame. 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. In the absence of applied torques, one obtains the Euler top. When the torques are due to gravity, there are special cases when the motion of the top is integrable. In an inertial frame of reference (subscripted "in"), Euler's second law states that the time derivative of the angular momentum L equals the applied torque: For point particles such that the internal forces are central forces, this may be derived using Newton's second law. For a rigid body, one has the relation between angular momentum and the moment of inertia Iin given as In the inertial frame, the differential equation is not always helpful in solving for the motion of a general rotating rigid body, as both Iin and ω can change during the motion. One may instead change to a coordinate frame fixed in the rotating body, in which the moment of inertia tensor is constant. Using a reference frame such as that at the center of mass, the frame's position drops out of the equations. In any rotating reference frame, the time derivative must be replaced so that the equation becomes and so the cross product arises, see time derivative in rotating reference frame. The vector components of the torque in the inertial and the rotating frames are related by where is the rotation tensor (not rotation matrix), an orthogonal tensor related to the angular velocity vector by for any vector u.