(untitled) (Communications in Mathematical Research, 2023)
As an application of the theoretical results, in this paper, we study the symmetric reduction and Hamilton-Jacobi theory for the underwater vehicle with two internal rotors as a regular point reducible RCH system, in the cases 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 Lie group. Next, we derive precisely the geometric constraint conditions of the reduced symplectic forms for the dynamical vector fields of the regular point reducible controlled underwater vehicle-rotor system, that is, the two types of Hamilton-Jacobi equations for the reduced controlled underwater vehicle-rotor system, by calculations in detail. The work reveals the deeply internal relationships of the geometrical structures of the phase spaces, the dynamical vector fields and the controls of the system.