This paper extends the theory of controlled Hamiltonian systems with symmetries due to [23,9,10,6,7,11] to the case of non-abelian symmetry groups $ G $ and semi-direct product configuration spaces. The notion of symmetry actuating forces is introduced and it is shown, that Hamiltonian systems subject to such forces permit a conservation law, which arises as a controlled perturbation of the $ G $-momentum map. Necessary and sufficient matching conditions are given to relate the closed-loop dynamics, associated to the forced Hamiltonian system, to an unforced Hamiltonian system. These matching conditions are then applied to general Lie-Poisson systems, to the example of ideal charged fluids in the presence of an external magnetic field ([20]), and to the satellite with a rotor example ([9,10]).

中文翻译：

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对称致动闭环哈密顿系统

本文扩展了由于[23，9，10，6，7，11]对于非阿贝尔对称群$ G $和半直接乘积配置空间。引入了对称致动力的概念，并且表明，受这种力作用的哈密顿系统允许守恒定律，该守恒定律是由G-动量图的受控扰动引起的。给出了必要和充分的匹配条件，以将与强制汉密尔顿系统相关的闭环动力学与非强制汉密尔顿系统相关联。然后将这些匹配条件应用于一般的Lie-Poisson系统，并在存在外部磁场（[[20]），以及带有转子示例（[9，10]）。