Phy 340 class notes for Fall, 1999

Aug 25	Sep 8	Oct 4	Nov 1	Phy 340 page
Aug 30	Sep 13	Oct 11	Nov 8
	Sep 20	Oct 18	Nov 15
	Sep 27	Oct 25	Nov 22
			Nov 29

Phy 340, Week 15

This theorem has as a consequence that fact that the components of the inertia tensor form a symmetric matrix in the usual orthonormal basis. Show the application to the symmetry of the components of the tensor in long-hand notation, then re-derive it in the index notation commonly used for this manipulation.

Under what circumstance is the angular velocity vector parallel to the angular momentum vector? Translate this into a mathematical equation, and the simplest approach is to say that one vector is a constant times the other vector: L = l w. The number l is called an "eigenvalue." Write this in terms of the inertia tensor, and you have I( w ) = l w.

In order to compute with this, it's useful to translate this into components. When you write it out completely, it is

One solution to this set is of course, w_x = w_y = w_z = 0, but that means nothing is happening. In order to get a more interesting, non-zero solution for the w's, you require that the determinant of the coefficients is zero. This is a cubic equation for l. Once you've found a root to this cubic, you then go back and find the corresponding w-components that are now going to exist. These are the components of the "eigenvector" of the tensor.

From the same identity proved at the start of today, I also proved the orthogonality of the eigenvectors corresponding to different eigenvalues.

1 Dec More inertia tensors. Compute in detail the components of one such tensor, and find its eigenvectors and eigenvalues. If I choose the set of eigenvectors as a basis, what are the components of the tensor in this basis? The result is a diagonal matrix.

3 Dec What are the dynamics of a rotating body? t = dL/dt. If L is expressed in terms of the inertia tensor and the angular velocity, then this is complicated by the fact that the tensor is itself a function of time. As the body rotates, all the components of I change. Unlessyou choose a coordinate system that is fixed in the rotating body. We've already worked out the transformation to a rotating system, so apply it to this case. Now the advantage of working in the basis of eigenvectors is apparent.

Work out a couple of examples. With w fixed, what is the condition for zero torque? It is that two out of the three components of w are zero. In this basis, that makes it an eigenvector.

What if there is no torque? Perhaps (to a good approximation) the Earth. If it is axially symmetric, then the resulting equations are linear and easily solved. They predict a slow precession of w around the axis. (Period around 400 days, the Chandler wobble.)

Discuss, but don't solve the stability of motion around an axis. It has practical applications.