Bifurcations of MacLaurin spheroids. A Hamiltonian perspective

TL;DR

This paper applies Hamiltonian bifurcation theory to analyze the bifurcations of MacLaurin spheroids, revealing three types of bifurcations consistent with Chandrasekhar's linearized hydrodynamic results.

Contribution

It introduces a Hamiltonian perspective to classify bifurcations of MacLaurin spheroids, providing a new theoretical framework for understanding their stability and transitions.

Findings

All bifurcations are into three types: I, S, and adjoint S ellipsoids.

Results agree with Chandrasekhar's linearized hydrodynamic conditions.

The Hamiltonian approach offers a systematic way to study fluid body bifurcations.

Abstract

Dirichlet's problem for the dynamics of fluid bodies with ellipsoidal shape can be formulated as a Hamiltonian system invariant under the action of a symmetry Lie group. I apply methods from Hamiltonian bifurcation theory to the study of the branch of solutions known as MacLaurin spheroids. I show that all its bifurcations are into three types named I, $S$ and adjoint $S$ ellipsoids in agreement with previous necessary conditions obtained by Chandrasekhar by linearizing the hydrodynamic equations.