Analytical integration of a generalized Euler-Poinsot problem: applications

R. Molina  y  A. Vigueras
Departamento de Matematica Aplicada y Estad EDstica
Escuela Politecnica Superior de Cartagena, Universidad de Murcia
C/ Paseo Alfonso XIII, 52.  30203 Cartagena (Murcia), Spain

- We consider the case of the free motion of a stationary gyrostat
about a fixed point O, belonging to its rigid part. First, we introduce
the Serret-Andoyer canonical variables for this problem and study all
possible solutions in the phase plane. Then, we analytically integrate 
a generalized Euler-Poinsot problem for a gyrostat whose first two
components of the gyrostatic momentum are null. The obtained solutions
are expressed in terms of elliptic functions and integrals, and they are
just the same as those for rigid bodies if a specific constant is
annulled.
Finally, two applications of these solutions are proposed:
 1) for obtaining the action-angle variables of this problem, and
 2) to the problem of the rotation of the Earth, using a triaxial
gyrostat as a model, the zero order for the Hamiltonian of the
perturbed problem is the Hamiltonian of a generalized Euler-Poinsot
problem.

Key words: Dynamics of rigid bodies and gyrostats, analogous case
         to that of Euler-Poinsot, analytic solutions and applications.