1. Euler Rotation

2. Angular Momentum  The angular momentum J is defined in terms of the inertia tensor and angular velocity. All rotations included  The angular momentum need not be collinear with the angular velocity. Not along principal axis Not at center of mass J r p

3. Torque  Torque N causes a change in angular momentum. Rotational second lawRotational second law  Use the body frame for a constant inertia tensor. Motion in accelerated frameMotion in accelerated frame

4. Euler Equations  Select the body coordinates to match the principal axes. Three moments of inertiaThree moments of inertia Simplified angular momentum termsSimplified angular momentum terms  Redo the torque equations.  These are Euler’s equations of motion.

5. Dumbbell  The principal axes are along and perpendicular to the rod.  Measure change in angular momentum. J l

6. Euler Angles  A rotation matrix can be described with three free parameters. Select three separate rotations about body axesSelect three separate rotations about body axes 1) Rotation of  about e 3 axis. 2) Rotation of  about e 1 axis. 3) Rotation of  about e 3 axis.  These are the Euler angles. e1e1 e2e2 e3e3   

7. Euler Matrices  Any vector z can be rotated though the Euler angles.  The equivalent matrix operation is the product of three separate operations.

8. Full Rotation  Any rotation may be expressed with the three angles. next