Euler's Equations (rigid Body Dynamics) - Motivation and Derivation

Motivation and Derivation

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 I_in 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 I_in 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

where M_k, I_k and ω_k are as above.

In a rotating reference frame, the time derivative must be replaced with (see time derivative in rotating reference frame)

where the subscript "rot" indicates that it is taken in the rotating reference frame. The expressions for the torque in the rotating and inertial frames are related by

where Q is the rotation tensor, an orthogonal tensor related to the angular velocity vector by

for any vector v.

In general, L = I·ω is substituted and the time derivatives are taken realizing that the inertia tensor, and so also the principal moments, do not depend on time. This leads to the general vector form of Euler's equations

If principal axis rotation

is substituted, and then taking the cross product and using the fact that the principal moments do not change with time, we arrive at the Euler equations in components at the beginning of the article.