# Motions about a fixed point by hypergeometric functions: new non-complex analytical solutions and integration of the herpolhode

## TL;DR

This paper presents new analytical solutions for the motion of a rigid body about a fixed point using hypergeometric functions, including explicit formulas for Euler angles, precession, herpolhode trajectories, and viscous dissipation effects.

## Contribution

It introduces novel non-complex analytical solutions for four classical rigid body problems, utilizing hypergeometric functions and extending previous treatments with explicit formulas and dissipation analysis.

## Key findings

- Explicit formulas for Euler angles using hypergeometric functions.
- New solutions for the herpolhode trajectory.
- Analysis of viscous dissipation effects on rigid body motion.

## 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 $-1/2$.