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. 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. Derivation 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 rotating and the inertial frames are related by where Q is the rotation tensor (not rotation matrix), an orthogonal tensor related to the angular velocity vector by for any vector u. Now is substituted and the time derivatives are taken in the rotating frame, while realizing that the particle positions and the inertia tensor does not depend on time. This leads to the general vector form of Euler's equations which are valid in such a frame The equations are also derived from Newton's laws in the discussion of the resultant torque. Principal axes form When choosing a frame so that its axes are aligned with the principal axes of the inertia tensor, its component matrix is diagonal, which further