$G$-isotropy of $T$-relative equilibria within manifolds tangent to spaces with linearly independent weights

TL;DR

This paper analyzes the local structure and isotropy properties of relative equilibria in Hamiltonian systems with symmetry, focusing on manifolds tangent to spaces with linearly independent weights and their stratification by isotropy types.

Contribution

It characterizes the isotropy groups of manifolds of $T$-relative equilibria and shows their stratification by isotropy type, extending understanding of symmetry in Hamiltonian systems.

Findings

Manifolds of $T$-relative equilibria are locally diffeomorphic to their tangent spaces, preserving isotropy groups.

The $G$-orbit of these manifolds is stratified by isotropy type, with explicit dimension formulas.

Examples illustrate points with same $T$-isotropy but different $G$-isotropy.

Abstract

We investigate the generic local structure of relative equilibria in Hamiltonian systems with symmetry $G$ near a completely symmetric equilibrium, where $G$ is compact and connected. Fix a maximal torus $T \subset G$ and identify the equilibrium with the origin within a symplectic representation of $G$. By a previous result, generically, for each $\xi \in \mathfrak t$ such that $V_0:= \ker \mathrm d^2(h-\mathbf J^\xi)(0)$ has linearly independent weights, there is a manifold tangent to $V_0$ that consists of relative equilibria with generators in $\mathfrak t$. Here we determine their isotropy with respect to $G$. The main result asserts that for each of these manifolds of $T$-relative equilibria, there is a local diffeomorphism to its tangent space at $0$ that preserves the isotropy groups. We will then deduce that the $G$-orbit of the union of these manifolds is stratified by isotropy type. The stratum given by the relative equilibria of type $(H)$ has the dimension $\dim G -\dim H + \dim (\mathfrak t')^L$, where $\mathfrak t' \subset \mathfrak t$ is the orthogonal complement of $\mathfrak h \cap \mathfrak t$ and $L$ is the minimal adjoint isotropy subgroup of an element of $\mathfrak t$ with $H \subset L$. In the end, we consider some examples of these manifolds of $T$-relative equilibria that contain points with the same isotropy type with respect to the $T$-action but different isotropy type with respect to $G$.