ANALYTICAL INTEGRATION OF A GENERALIZED EULER-POINSOT PROBLEM: APPLICATIONS

R. Molina  y  A. Vigueras=20
Departamento de Matem=E1tica Aplicada y Estad=EDstica
Escuela Polit=E9cnica Superior de Cartagena, Universidad de Murcia
C/ Paseo Alfonso XIII, 52.  30203 Cartagena (Murcia), SPAIN=20

Abstract.- 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.