Non integrability of a self-gravitating Riemann liquid ellipsoid

TL;DR

This paper proves that the dynamics of a triaxial Riemann liquid ellipsoid without angular momentum are non-integrable, showing no additional meromorphic first integrals exist, even at fixed energy levels.

Contribution

It provides a rigorous proof of non-integrability for the self-gravitating liquid ellipsoid system, extending understanding of its complex dynamical behavior.

Findings

No additional meromorphic first integrals exist

System is non-integrable even on fixed energy hypersurfaces

Proves non-integrability rigorously for the model

Abstract

We prove that the motion of a triaxial Riemann ellipsoid of homogeneous liquid without angular momentum does not possess an additional first integral which is meromorphic in position, impulsions, and the elliptic functions which appear in the potential, and thus is not integrable. We prove moreover that this system is not integrable even on a fixed energy level hypersurface.