1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

392 J.E. MARSDEN, MECHANICS, DYNAMICS, AND SYMMETRY

Consider a carrier rigid body with a rotor aligned along the third principal axis.
The rotor spins under the influence of a torque u , as in figure 4.3.

rigid carrier

Figure 4.3. A rigid body with a rotor a ligned on the long axis.

The equations of motion are

rr = 11 x n, i = u

where Ii > h > h are the carrier moments of inertia, J1 = h and h are the rotor

moments of inertia, n = (!1 1 ,!1 2 ,!1 3 ) are the carrier angular velocity, and a is the

relative angle of the rotor. The body angular momenta are given by

.A1D1; 112 = .A2D2

,\3!1 3 + J3&.; l3 = h(!13 + &.)

where ,\i =Ii+];.

The equations in components read

u.

If u = 0, then l3 is a constant of motion and the remaining equations are

Hamiltonian (Lie-Poisson) with

H _ ~ (Ili 11 ~ (113 - l3)

2

) ~l 2

- 2 .A 1 + .A 2 + I 3 + 2 3 ·

We use the feedback control law:

u = k (_!_ -!__) 11111 2,

.A 2 .A1