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ABSTRACT: In this paper, our goal is to study the regular reduction theory of regular
controlled Hamiltonian (RCH) systems with symplectic structure and symmetry,
and this reduction is an extension of regular symplectic reduction theory of
Hamiltonian systems under regular controlled Hamiltonian equivalence
conditions. Thus, in order to describe uniformly RCH systems defined on a
cotangent bundle and on the regular reduced spaces, we first define a kind of
RCH systems on a symplectic fiber bundle. Then introduce regular point and
regular orbit reducible RCH systems with symmetry by using momentum map and the
associated reduced symplectic forms. Moreover, we give regular point and
regular orbit reduction theorems for RCH systems to explain the relationships
between RpCH-equivalence, RoCH-equivalence for reducible RCH systems with
symmetry and RCH-equivalence for associated reduced RCH systems. Finally, as an
application we regard rigid body and heavy top as well as them with internal
rotors as the regular point reducible RCH systems on the rotation group
$\textmd{SO}(3)$ and on the Euclidean group $\textmd{SE}(3)$, respectively, and
discuss their RCH-equivalence. We also describe the RCH system and
RCH-equivalence from the viewpoint of port Hamiltonian system with a symplectic
structure.
02/2012;
