与具有线性无关权的空间相切的流形中$T$ -相对平衡的$G$ -各向同性
$G$-isotropy of $T$-relative equilibria within manifolds tangent to spaces with linearly independent weights
摘要：
研究了具有对称性$G$的Hamilton系统在完全对称平衡点附近的相对平衡点的一般局部结构，其中$G$是紧的和连通的。固定一个最大环面$T \subset G$，并在$G$的辛表示中确定原点的平衡。根据前面的结果，一般地，对于每个$ξ\in \mathfrak t$，使得$V_0:= \ker \mathrm d^2(h-\mathbf J^ξ)(0)$具有线性无关权重，存在与$V_0$相切的流形，该流形由与$\mathfrak t$中的生成元的相对平衡组成。在这里，我们确定他们的各向同性相对于$G$。主要结果断言，对于每个这些流形的$T$ -相对平衡，有一个局部的切空间在$0$，保持各向同性群。然后，我们将推出的$G$ -轨道的联合这些流形是分层的各向同性型。由$(H)$型的相对平衡给出的层具有维数$\dim G -\dim H + \dim (\mathfrak t')^L$，其中$\mathfrak t' \subset \mathfrak t$是$\mathfrak h \cap \mathfrak t$的正交补，而$L$是$\mathfrak t$的元素与$H \subset L$的最小伴随各向同性子群。最后，我们考虑了一些例子，这些流形的$T$ -相对平衡，包含点具有相同的各向同性类型的$T$ -行动，但不同的各向同性类型的$G$。
英文摘要：
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 $ξ\in \mathfrak t$ such that $V_0:= \ker \mathrm d^2(h-\mathbf J^ξ)(0)$ 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 $G$.
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