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

Motions about a fixed point by hypergeometric functions: new non-complex analytical solutions and... The present study highlights the dynamics of a body moving about a fixed point and provides analytical closed form solutions. Firstly, for the symmetrical heavy body, that is the Lagrange–Poisson case, we compute the second (precession, $$\psi$$ ψ ) and third (spin, $$\varphi$$ φ ) Euler angles in explicit and real form by means of multiple hypergeometric (Lauricella) functions. Secondly, releasing the weight assumption but adding the complication of the asymmetry, by means of elliptic integrals of third kind, we provide the precession angle $$\psi$$ ψ completing the treatment of the Euler–Poinsot case. Thirdly, by integrating the relevant differential equation, we reach the finite polar equation of a special motion trajectory named the herpolhode. Finally, we keep the symmetry of the first problem, but without weight, and take into account a viscous dissipation. The use of motion first integrals—adopted for the first two problems—is no longer practicable in this situation; therefore, the Euler equations, faced directly, are driving to particular occurrences of Bessel functions of order $$-\,1/2$$ - 1 / 2 . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Celestial Mechanics and Dynamical Astronomy Springer Journals

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

, Volume 130 (6) – Jun 1, 2018
34 pages

Publisher
Springer Netherlands
Subject
Physics; Astrophysics and Astroparticles; Dynamical Systems and Ergodic Theory; Aerospace Technology and Astronautics; Geophysics/Geodesy; Classical Mechanics
ISSN
0923-2958
eISSN
1572-9478
D.O.I.
10.1007/s10569-018-9837-5
Publisher site
See Article on Publisher Site

### Abstract

The present study highlights the dynamics of a body moving about a fixed point and provides analytical closed form solutions. Firstly, for the symmetrical heavy body, that is the Lagrange–Poisson case, we compute the second (precession, $$\psi$$ ψ ) and third (spin, $$\varphi$$ φ ) Euler angles in explicit and real form by means of multiple hypergeometric (Lauricella) functions. Secondly, releasing the weight assumption but adding the complication of the asymmetry, by means of elliptic integrals of third kind, we provide the precession angle $$\psi$$ ψ completing the treatment of the Euler–Poinsot case. Thirdly, by integrating the relevant differential equation, we reach the finite polar equation of a special motion trajectory named the herpolhode. Finally, we keep the symmetry of the first problem, but without weight, and take into account a viscous dissipation. The use of motion first integrals—adopted for the first two problems—is no longer practicable in this situation; therefore, the Euler equations, faced directly, are driving to particular occurrences of Bessel functions of order $$-\,1/2$$ - 1 / 2 .

### Journal

Celestial Mechanics and Dynamical AstronomySpringer Journals

Published: Jun 1, 2018

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