Symmetric Reduction and Hamilton-Jacobi Equations of the Controlled Underwater Vehicle-Rotor System

As an application of the theoretical results,in this paper,we study the symmetric reduction and Hamilton-Jacobi theory for the underwater ve-hicle 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 sym-metry 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.These work reveal the deeply internal relationships of the geometrical structures of the phase spaces,the dynamical vector fields and the controls of the system.