Title: Motions about a fixed point by hypergeometric functions: new non-complex analytical solutions and integration of the herpolhode,
Abstract: We study four problems in the dynamics of a body moving about a fixed point,
providing a non-complex, analytical solution for all of them. For the first
two, we will work on the motion first integrals. For the symmetrical heavy
body, that is the Lagrange-Poisson case, we compute the second and third Euler
angles in explicit and real forms by means of multiple hypergeometric functions
(Lauricella, functions). Releasing the weight load but adding the complication
of the asymmetry, by means of elliptic integrals of third kind, we provide the
precession angle completing some previous treatments of the Euler-Poinsot case.
Integrating then the relevant differential equation, we reach the finite polar
equation of a special trajectory named the {\it herpolhode}. In the last
problem we keep the symmetry of the first problem, but without the weight, and
take into account a viscous dissipation. The approach of first integrals is no
longer practicable in this situation and the Euler equations are faced directly
leading to dumped goniometric functions obtained as particular occurrences of
Bessel functions of order $-1/2$.