论文标题
$ g $ - $ t $ t $ - 偏向平衡的歧管与线性独立权重的空间相切
$G$-isotropy of $T$-relative equilibria within manifolds tangent to spaces with linearly independent weights
论文作者
论文摘要
我们研究了在完全对称的平衡附近的哈密顿系统中相对平衡的通用局部结构，其中$ g $是紧凑且连接的。修复最大圆环$ t \子集g $，并在$ g $的符号表示内确定与原点的平衡。总体而言，对于先前的结果，对于\ mathfrak t $中的每个$ξ\ t $，以至于$ v_0：= \ ker \ mathrm d^2（h- \ mathbf j^ξ）（0）$具有线性独立的权重，与$ v_0 $相对均与$ v_0 $相对的$ v_0 $，由$ V_0与$ \ mathfrak t $ \ \ \ \ \ \ \ \ \ \ mathers组成。在这里，我们确定它们相对于$ g $。主要结果断言，对于$ t $依赖的平衡的每种流形，其切线空间$ 0 $都存在局部差异性，可以保留各向同性群。然后，我们将推断出这些歧管结合的$ g $ - 轨道是通过各向同性类型分层的。 $（h）$的相对平衡给出的层面具有尺寸$ \ dim g- \ dim dim h + \ dim（\ mathfrak t'）^l $，其中$ \ mathfrak t'\ subset \ subset \ mathfrak t $是$ \ mathfrak H \ cap \ cap \ mathfrak is y Mathfrak T $ lim y Mathfrak T $ lim lim lim y Mathfrak is y Minim of Math us of unthogona $ \ mathfrak t $带有$ h \ subset l $。最后，我们考虑了$ t $相关的这些流形的一些示例，这些示例包含相对于$ t $ Action，但相对于$ g $不同的各向同性类型的点相同。
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 $G$.
