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 3D 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.
Starting from Euler's second law, in an inertial frame of reference (subscripted "in"), the time derivative of the angular momentum L equals the applied torque
where Iin is the moment of inertia tensor calculated in the inertial frame. Although this law is universally true, it is not always helpful in solving for the motion of a general rotating rigid body, since both Iin and ω can change during the motion.
Therefore, we change to a coordinate frame fixed in the rotating body, and chosen so that its axes are aligned with the principal axes of the moment of inertia tensor. In this frame, at least the moment of inertia tensor is constant (and diagonal), which simplifies calculations. As described in the moment of inertia, the angular momentum L can be written
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