Before we can write down the rotational equations of motion we have to define principle axes and moments of inertia. A principal axis of inertia is defined obviously by a direction. Let's say u is that direction while u is a unit vector along a principal axis, if I times u is parallel to u. There's a theorem that says that you can always find three such independent principle axes. In other words there are three independent axes such that I times the unit vector along that axis will give you a vector that's parallel to that axis. The moment of inertia is essentially the scaling term. So if I take I times u and it defers from U by a scale of factor, that scale of factor is the moment of inertia. These moments of inertia are called principal moments of inertia. Here's a simple example that illustrates what the moments of inertia tell you. So in this cartoon, you have a rigid frame, A you have a parallel plate that's spinning about a vertical axis, but there are two different configurations. On the right side, the axis is perpendicular to the plate. In fact this configuration is symmetric. On the left side, the configuration is symmetric, but the axis is not perpendicular to the plate. On the right side, if you compute the angular momentum. You'll find that the angular momentum vector and the angular velocity vector are parallel. On the left side, they are not. And this is because the axis of rotation does not coincide with any of the principal axes. Now we're ready to deal with Euler's Equations which will tell us the rotational equations of motion. Again it comes down to this basic observation. The rate of change of angular momentum is equal to the net moment applied to the rigid body. We take C, the center of mass as the origin for all our calculations. In this picture, B1, B2 and B3 are a set of body fixed unit vectors that define a body fixed frame. We'll now insist that these vectors point along principal axes and we'll write our angular velocity as linear combinations of b1, b2, and b3, and the components are omega 1, omega 2, and omega 3. Once again remember, the two key aspects to what we are doing, one is that we're taking c, the center of mass of the origin. Second we're insisting that the body fixed frame in which we'll do our calculations will be along the three principal axes. If you take the basic equation on the top right, you can write the left hand side by breaking it up into two terms. The first term involves the derivative in a body fixed frame. The second term and while it's a correction factor, that correction factor's a vector that takes into account the fact that the differentiation is done in the body fixed frame. This correction factor is simply the angular velocity of the moving body fixed frame crossed with the angular momentum. So this correction factor as I paraphrased it, is actually a well-known fact in mechanics. Anytime you differentiate a vector in a moving frame, it's derivative is different from the derivative in a fixed frame. The difference is obtained by simply factoring in the cross product of the angular velocity with that vector. But first term on the left hand side can be written in terms of inertia matrix times the angular velocity vector. Because we have chosen principle axes, it turns out that the off diagonal elements in the inertia tensor are zero. So therefore the first term which involves I times omega, the inertia tensor times the angular velocity vector consists of three terms. The diagonal terms I11, I22, I33, which multiply with omega1, omega2, and omega3, yield three terms that you see on the right hand side. The second term on the left hand side of the equation immediately above that which is omega cross H can be written again in component form. And that gives you the matrix equation that you see on top. These are Euler's Equations of Motion. You can see they're quite compact. The first term Is essentially the derivative of the angular momentum in a body fixed frame. The second term is the correction. The third term, the term on the right hand side, is the net moment. In what follows we will use p,q and r to denote the components of the angular velocity vector along b1, b2, and b3.
