On a new type of solving procedure for Euler-Poisson equations (rigid body rotation over the fixed point)

In this paper, we proceed to develop a new approach which was formulated first in Ershkov (Acta Mech 228(7):2719-2723, 2017) for solving Poisson equations: a new type of the solving procedure for Euler-Poisson equations (rigid body rotation over the fixed point) is suggested in the current research. Meanwhile, the Euler-Poisson system of equations has been successfully explored for the existence of analytical solutions. As the main result, a new ansatz is suggested for solving Euler-Poisson equations: the Euler-Poisson equations are reduced to a system of three nonlinear ordinary differential equations of first order in regard to three functions Omega(i) (i = 1,2,3); the proper elegant approximate solution has been obtained as a set of quasi-periodic cycles via re-inversing the proper elliptical integral. So the system of Euler-Poisson equations is proved to have analytical solutions (in quadratures) only in classical simplifying cases: (1) Lagrange's case, or (2) Kovalevskaya's case or (3) Euler's case or other well-known but particular cases.