# American Institute of Mathematical Sciences

December  2020, 12(4): 641-669. doi: 10.3934/jgm.2020030

## Symmetry actuated closed-loop Hamiltonian systems

 Österreichische Finanzmarktaufsicht (FMA), Otto-Wagner Platz 5, A-1090 Vienna, Austria

Received  December 2019 Published  December 2020 Early access  November 2020

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]).

Citation: Simon Hochgerner. Symmetry actuated closed-loop Hamiltonian systems. Journal of Geometric Mechanics, 2020, 12 (4) : 641-669. doi: 10.3934/jgm.2020030
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