where Iin is the moment of inertiatensor 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
In general, L = Iω 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.
For the RHSs equal to zero there are non-trivial solutions: torque-free precession. Notice that since I is constant (because the inertia tensor is a 3×3 diagonal matrix (see the previous section), because we work in the intrinsic frame, or because the torque is driving the rotation around the same axis so that I is not changing) then we may write
However, if I is not constant in the external reference frame (i.e. the body is moving and its inertia tensor is not constantly diagonal) then we cannot take the I outside the derivative. In this case we will have torque-free precession, in such a way that I(t) and ω(t) change together so that their derivative is zero. This motion can be visualized by Poinsot's construction.
It is also possible to use these equations if the axes in which
is described are not connected to the body. Then ω should be replaced with the rotation of the axes instead of the rotation of the body. It is, however, still required that the chosen axes are still principal axes of inertia. This form of the Euler equations is useful for rotation-symmetric objects that allow some of the principal axes of rotation to be chosen freely.
C. A. Truesdell, III (1991) A First Course in Rational Continuum Mechanics. Vol. 1: General Concepts, 2nd ed., Academic Press. ISBN0-12-701300-8. Sects. I.8-10.
C. A. Truesdell, III and R. A. Toupin (1960) The Classical Field Theories, in S. Flügge (ed.) Encyclopedia of Physics. Vol. III/1: Principles of Classical Mechanics and Field Theory, Springer-Verlag. Sects. 166-168, 196-197, and 294.