Relative equilibria of Hamiltonian systems with symmetry: Linearization, smoothness, and drift

Relative Equilibria and Conserved Quantities in Symmetric Hamiltonian Systems

“…In an important paper [60], George Patrick investigates the structure of the set of relative equilibria as quoted in the following theorem. He also studied the nearby dynamics and introduced the notion of drift around relative equilibria in terms of the linearized vector field there, but we will not be describing that aspect here.…”

“…Theorem 5.1 (Patrick [60]) Assume G is compact, G p is finite and G ξ ∩ G µ is a maximal torus. Then in a neighbourhood of p the set of relative equilibria forms a smooth symplectic submanifold of P of dimension dim(G)+ rank(G).…”

Stability by KAM Confinement of Certain Wild, Nongeneric Relative Equilibria of Underwater Vehicles with Coincident Centers of Mass and Buoyancy

“…The appearance of the nilpotent part of the linearization is the foundational element of [6]. The value of κ e coincides with the derivative dξ αe e /dμ αe e from (1.5), as predicted by the general theory.…”

“…These equilibria are fixed points of the action of SO (2), and the analysis requires a transparent extension of the normal form in [6] to equilibria which have SO (2) …”

“…The argument proceeds by adapting and incrementally refining, to the order required for the stability analysis, the normal form for relative equilibria developed in [6].…”