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

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.

15 pages

References in corpus (7)

• Lagrangian and Hamiltonian formulations of asymmetric rigid body, considered as a constrained system
• Euler-Poisson equations of a dancing spinning top, integrability and examples of analytical solutions
• An asymmetrical body: example of analytical solution for the rotation matrix in elementary functions and Dzhanibekov effect
• General solution to the Euler-Poisson equations of a free Lagrange top directly for the rotation matrix
• Rotation matrix of a charged symmetrical body: one-parameter family of solutions in elementary functions
• Has the problem of the motion of a heavy symmetric top been solved in quadratures?
• Dynamics on a submanifold: intermediate formalism versus Hamiltonian reduction of Dirac bracket, and integrability