General solution to Euler-Poisson equations of a free symmetric body by direct summation of power series
Guilherme Corrêa Silva
TL;DR
The paper tackles the Euler-Poisson equations for a free symmetric rigid body and derives a closed-form, Euler-angle-free solution by summing the analytic power-series representations guaranteed by the Cauchy-Kovalevskaya theorem. By exploiting integrals of motion and an operator-exponential framework, the authors obtain explicit, elementary expressions for the angular-velocity vector $\boldsymbol{\Omega}(t)$ and the rotation matrix $R_{ij}(t)$, including the regular precession characterized by frequencies $k$ and $\phi$. The results reproduce the known rotation dynamics, such as $\Omega_1=(m_2/I_2)\sin\phi t$, $\Omega_2=(m_2/I_2)\cos\phi t$, and $\Omega_3=m_3/I_3$, and provide a direct, angle-free derivation of $R(t)$ via two complementary summation approaches. This work offers a new analytic perspective on rigid-body dynamics and presents a general technique for summing CK-series in autonomous systems.
Abstract
Euler-Poisson equations describe the temporal evolution of a rigid body's orientation through the rotation matrix and angular velocity components, governed by first-order differential equations. According to the Cauchy-Kovalevskaya theorem, these equations can be solved by expressing their solutions as power series in the evolution parameter. In this work, we derive the sum of these series for the case of a free symmetric rigid body. By using the integrals of motion and directly summing the terms of these series, we obtain the general solution to the Euler-Poisson equations for a free symmetric body in terms of elementary functions. This method circumvents the need for standard parametrizations like Euler angles, allowing for a direct, closed-form solution. The results are consistent with previous studies, offering a new perspective on solving the Euler-Poisson equations.