The dynamics around stable and unstable Hamiltonian relative equilibria

Bifurcation of relative equilibria in mechanical systems with symmetry

The relative equilibria of a symmetric Hamiltonian dynamical system are the critical points of the so-called augmented Hamiltonian. The underlying geometric structure of the system is used to decompose the critical point equations and construct a collection of implicitly defined functions and reduced equations describing the set of relative equilibria in a neighborhood of a given relative equilibrium. The structure of the reduced equations is studied in a few relevant situations. In particular, a persistence result of Lerman and Singer [LS98] is generalized to the framework of Abelian proper a…

缩放对称性与皮亚诺 manifold 上的共形相对平衡，及其在李-皮亚诺系统中的应用

Scaling Symmetries and Conformal Relative Equilibria on Poisson Manifolds, with Applications to Lie--Poisson Systems

研究了具有缩放对称性的哈密顿系统在精确皮亚诺 manifold 上的共形相对平衡，通过引入共形皮亚诺作用和共形动量映射，建立了一种扩展的哈密顿公式。在非退化情况下，恢复了最近为精确辛情况发展的条件。

On relative normal modes

We generalize the Weinstein-Moser theorem on the existence of nonlinear normal modes near an equilibrium in a Hamiltonian system to a theorem on the existence of relative perodic orbits near a relative equilibrium in a Hamiltonian system with continuous symmetries. In particular we prove that under appropriate hypotheses there exist relative periodic orbits near relative equilibria even when these relative equilibria are singular points of the corresponding moment map, i.e. when the reduced spaces are singular.

Relative equilibria at singular points of the moment map

We prove a criterion for stability of relative equilibria in symmetric Hamiltonian systems at singular points of the momentum map. This generalizes a theorem of G.W. Patrick. The method of the proof is also useful in studying the bifurcation of relative equilibria.

缩放对称性的相对平衡与中心配置

Relative Equilibria for Scaling Symmetries and Central Configurations

本文探讨了缩放对称性在辛几何框架下的作用，引入了共形辛映射和共形动量映射，推导了缩放对称性的相对平衡方程，并应用于牛顿n体问题。

Nonlinear Stability of Relative Equilibria and Isomorphic Vector Fields

We present applications of the notion of isomorphic vector fields to the study of nonlinear stability of relative equilibria. Isomorphic vector fields were introduced by Hepworth [Theory Appl. Categ. 22 (2009), 542-587] in his study of vector fields on differentiable stacks. Here we argue in favor of the usefulness of replacing an equivariant vector field by an isomorphic one to study nonlinear stability of relative equilibria. In particular, we use this idea to obtain a criterion for nonlinear stability. As an application, we offer an alternative proof of Montaldi and Rodríguez-Olmos's criter…