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2020, Volume 12, Issue 4: 641-669. Doi: 10.3934/jgm.2020030

This issue Previous Article Lagrangian reduction of nonholonomic discrete mechanical systems by stages Next Article Erratum for "nonholonomic and constrained variational mechanics"

Symmetry actuated closed-loop Hamiltonian systems

Hochgerner Simon

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

Received: December 2019

Early access: November 2020

Published: December 2020

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Abstract

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]).

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Mathematics Subject Classification: 70G40, 10H14, 37K45, 76B75.

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