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 three-dimensional 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.

Motivation and derivation

Starting from Newton's second law, in an inertial frame of reference, 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, which simplifies calculations. As described in the moment of inertia, the angular momentum L can be written
where Mk, Ik and ωk are as above.
In a rotating reference frame, the time derivative must be replaced with
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 = 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.

Torque-free solutions

For the RHSs equal to zero there are non-trivial solutions: torque-free precession. Notice that since I is constant then we may write
where
However, if I is not constant in the external reference frame then we cannot take the I outside the derivative. In this case we will have torque-free precession, in such a way that I and ω change together so that their derivative is zero. This motion can be visualized by Poinsot's construction.

Generalizations

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.