1. 1 Class #28 Gyroscope demo. Angular momentum is real! Non-axi-symmetric rotations The inertia tensor Torques for off-axis rotations Diagonalization and eigenvalues Principal axes :02

2. 2 You thought you knew all about angular momentum…  P=mv … what could be more familiar?  Let’s just relax  Cross product  …and again :02

3. 3 Inertia Tensor  Inertia tensor -- single particle :02

4. 4 Inertia Tensor physical meaning :02 Given a pure rotation about z-axis, Izz gives the angular momentum that would occur along the z-axis. Ixz is angular momentum that occurs about x-axis due to a rotation about z-axis.

5. 5 Charles “Chuck” Hermite (1822-1901) :02 French mathematician who did brilliant work in many branches of mathematics, but was plagued by poor performance in exams as a student. However, on his own, he mastered Lagrange's memoir on the solution of numerical equations and Gauss's Disquisitiones Arithmeticae. He was forced to leave Ecole Polytechnique after one year when it was decided that his congenitally deformed right leg would not allow him to take a commission in the military, making him not worth their time.Lagrange'sGauss's In 1869, he became a professor at École Normale, and in 1870 at Sorbonne. All during his career, was generous in his help of young mathematicians. He showed that e was a transcendental number Hermite also discovered some of the properties of Hermitian matrices. etranscendental number Hermitian matrices

6. 6 Inertia Tensor is Hermitian Inertia tensor is symmetric Inertia tensor is Hermitian :02 Hermitian matrices have real eigenvalues If the eigenvalues differ, the eigenvectors are orthogonal

7. 7 Inertia Tensor :02  Derived all this for a single particle Below, for multiple particles

8. 8 Angular momentum of non-axi-symmetric object  Inclined bar-bell Bar length 2L :02 x y z

9. 9 Non-axisym. object II :60

10. 10 Non-axisym. object II :60  Let

11. 11 Non-axisym. object III :60

12. 12 Non-axisym. object IV :60

13. 13 But that’s not all!!  Principal Axes  The ones about which rotations do not produce torques  These are the eigenvectors of the inertia tensor  How do we find them? :02 x y z

14. 14 Eigenvalues and Eigenvectors :60 For arbitrary matrix A and vector x, if Ax=kx Where k=constant, Then x is an eigenvector of matrix A and ‘k’ is the eigenvalue for That vector. Not all matrices have non-zero eigenvectors They ONLY do if

15. 15 Diagonalization :60  Set phi=0  If is an eigenvector

16. 16 Diagonalization :60

17. 17 Problem L16-1 :30 x y z Given all the above, what are the principal axes (the eigenvectors) for this problem?