General solution to Euler–Poisson equations of a free symmetric body by direct summation of power series

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.