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Bifurcations of MacLaurin spheroids. A Hamiltonian perspective

Miguel Rodríguez-Olmos

TL;DR

This work addresses the problem of describing bifurcations from the MacLaurin family of Riemann ellipsoids within Dirichlet's gravitational problem by employing geometric Hamiltonian methods for relative equilibria. It develops and applies a framework based on the Energy–Momentum method, the symplectic normal space, and stabilizer analysis to track how branches of relative equilibria bifurcate from the MacLaurin spheroids. The main result is that three distinct bifurcation branches emerge: Type I ellipsoids, Type $S$ ellipsoids, and adjoint Type $S$ ellipsoids, with precise stability and symmetry conditions: Type I occurs for all $e\in(0,1)$ with trivial stabilizers, Type $S$ appears for $e<e_{\text{crit}}$ and yields Jacobi/Dedekind-like configurations, and adjoint Type $S$ occurs at $e=e_{\text{crit}}$ with a specific symmetry group. This Hamiltonian perspective aligns with Chandrasekhar's necessary conditions and clarifies the organization of Riemann ellipsoids near the MacLaurin branch, bridging classical results with modern geometric mechanics.

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.

Bifurcations of MacLaurin spheroids. A Hamiltonian perspective

TL;DR

This work addresses the problem of describing bifurcations from the MacLaurin family of Riemann ellipsoids within Dirichlet's gravitational problem by employing geometric Hamiltonian methods for relative equilibria. It develops and applies a framework based on the Energy–Momentum method, the symplectic normal space, and stabilizer analysis to track how branches of relative equilibria bifurcate from the MacLaurin spheroids. The main result is that three distinct bifurcation branches emerge: Type I ellipsoids, Type ellipsoids, and adjoint Type ellipsoids, with precise stability and symmetry conditions: Type I occurs for all with trivial stabilizers, Type appears for and yields Jacobi/Dedekind-like configurations, and adjoint Type occurs at with a specific symmetry group. This Hamiltonian perspective aligns with Chandrasekhar's necessary conditions and clarifies the organization of Riemann ellipsoids near the MacLaurin branch, bridging classical results with modern geometric mechanics.

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, and adjoint ellipsoids in agreement with previous necessary conditions obtained by Chandrasekhar by linearizing the hydrodynamic equations.
Paper Structure (14 sections, 1 theorem, 47 equations, 2 figures)

This paper contains 14 sections, 1 theorem, 47 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Hamiltonian bifurcations from parametrized branches of relative equilibria
  3. Bifurcations from the MacLaurin family
  4. Riemann's Theorem.
  5. Symmetries of Riemann ellipsoids and conditions on the possible bifurcations.
  6. Characterization of the MacLaurin branch.
  7. The symplectic normal space
  8. The action and momentum map.
  9. The stability form.
  10. Zero eigenvalues.
  11. Type I ellipsoids.
  12. Type $S$ ellipsoids.
  13. Type adjoint $S$ ellipsoids.
  14. Acknowledgments:

Key Result

Theorem 2.1

Let $z\in\mathcal{P}$ be a relative equilibrium with momentum $\mathbf{J}(z)=\mu$ satisfying $[\mathfrak{g}_\mu, \mathfrak{g}_\mu]=0$ and velocity $\xi\in\mathfrak{g}_\mu$ written as $\xi=\xi^\perp+\eta$ according to gmusplitting. Let $W\subseteq {\mathfrak{m}^*}^{G_z} \times \mathfrak{g}_z^{(G_z)_\ Then, for every $v\in \mathcal{N}$ close enough to the origin, there is a relative equilibrium $\wi

Figures (2)

  • Figure 1: Graph of $\hat{\mu}(e)$ in $(G\pi\rho_0)^\frac{1}{2}$ units along the MacLaurin family. It allows parametrizing the MacLaurin family by $e\in(0,1)$ or by $\hat{\mu}\in (0,\infty)$.
  • Figure 2: Graphs of $\eta_1^2(e)$ (top) and $\eta_2^2(e)$ (bottom) along the MacLaurin family expressed in $G\pi\rho_0$ units. Notice how $\eta_2$ becomes imaginary at $e_\mathrm{crit}\simeq 0.952887$, which is the point where the MacLaurin spheroid becomes unstable, while $\eta_1(e)$ is always real. There is no value of the eccentricity for which $\eta_1(e)=\eta_2(e)$.

Theorems & Definitions (1)

  • Theorem 2.1