Notes on approaches for solving the Euler-Poisson equations

TL;DR

This paper introduces a new analytical approach for solving the Euler-Poisson equations, reducing them to simpler forms and deriving approximate solutions, with exact solutions found in classical special cases.

Contribution

A novel method for solving Euler-Poisson equations is proposed, including a reduction to nonlinear ODEs and derivation of approximate solutions, extending analytical solution techniques.

Findings

Euler-Poisson equations can be reduced to 3 nonlinear ODEs.

Analytical solutions are obtainable in classical special cases.

Approximate solutions are derived via re-inversion of integrals.

Abstract

In this paper, we proceed to develop a new approach which was formulated first in Ershkov (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 way for presentation of the solution. As the main result, the new ansatz is suggested for solving Euler-Poisson equations: the Euler-Poisson equations are reduced to the system of 3 nonlinear ordinary differential equations of 1-st order in regard to 3 functions; the elegant approximate solution has been obtained via re-inversion of the proper analytical integral as a set of quasi-periodic cycles. So, the system of Euler-Poisson equations is proved to have the analytical solutions (in quadratures) only in classical simplifying cases: 1) Lagrange case, or 2) Kovalevskaya case or 3) Euler case or other well-known but particular cases.