报告题目：Regular Controlled Hamiltonian Systems and Hamilton-Jacobi Equations

报告时间：2023-12-06   10:00-12:00

报 告 人 ：王红  教授  南开大学

报告地点：理学院东北楼四楼学术报告厅

Abstract：A regular controlled Hamiltonian (RCH) system is a Hamiltonian system with external force and control. In general, an RCH system under the actions of external force and control is not Hamiltonian, however, it is a dynamical system closely related to a Hamiltonian system, and it can be explored and studied by extending the methods for external force and control in the study of Hamiltonian systems. In this report, from the viewpoint of completeness of Marsden-Weinstein reduction, we first give a definition of the RCH system, as well as a good expression of the dynamical vector field of the RCH system, such that we can describe the RCH-equivalence and symmetric reduction theory for RCH system. Secondly, we derive precisely the geometric constraint conditions of the canonical symplecticform for the dynamical vector field of the RCH system, that is, the Type I and Type II Hamilton-Jacobi equations. Thirdly, as an application of the theoretical results, 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. These researches reveal the deeply internal relationships of the geometrical structures of phase spaces, the dynamical vector fields and controls of the RCH system.