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

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 $\dim G-\dim H+\dim ({\mathfrak{t}}^{\prime }{)}^L$, where ${\mathfrak{t}}^{\prime }\subset \mathfrak{t}$ 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 $G$.

arXiv comment39 pages, 5 figures (tikZ)

References in corpus (1)

• The Transversal Relative Equilibria of a Hamiltonian System with Symmetry