Symmetric Reduction and Hamilton-Jacobi Equations of the Controlled Underwater Vehicle-Rotor System (Communications in Mathematical Research, 2023)
As an application of the theoretical results, in this paper, we studythe symmetric reduction and Hamilton-Jacobi theory for the underwater vehicle with two internal rotors as a regular point reducible RCH system, in thecases of coincident and non-coincident centers of the buoyancy and the gravity.At first, we give the regular point reduction and the two types of Hamilton-Jacobi equations for a regular controlled Hamiltonian (RCH) system with symmetry and a momentum map on the generalization of a semidirect product Liegroup. Next, we derive precisely the geometric constraint conditions of thereduced symplectic forms for the dynamical vector fields of the regular pointreducible controlled underwater vehicle-rotor system, that is, the two typesof Hamilton-Jacobi equations for the reduced controlled underwater vehicle-rotor system, by calculations in detail. The work reveals the deeply internalrelationships of the geometrical structures of the phase spaces, the dynamicalvector fields and the controls of the system.
Keywords: Underwater vehicle with internal rotors; regular controlled Hamiltonian system; coincident and non-coincident centers; regular point reduction; Hamilton-Jacobi equation.