HAMILTON-JACOBI EQUATIONS FOR A REGULAR CONTROLLED HAMILTONIAN SYSTEM AND ITS REDUCED SYSTEMS

作     者：王红 Hong WANG

作者机构：School of Mathematical Sciences and LPMCNankai UniversityTianjin 300071China 

出 版 物：《Acta Mathematica Scientia》 (数学物理学报（B辑英文版）)

年 卷 期：2023年第43卷第2期

页      面：855-906页

核心收录：

学科分类：07[理学] 070104[理学-应用数学] 0701[理学-数学] 

基　　金：partially supported by the Nankai University 985 Project the Key Laboratory of Pure Mathematics and Combinatorics,Ministry of Education,China the NSFC(11531011)。 

主　　题：regular controlled Hamiltonian system Hamilton-Jacobi equation regular point reduction regular orbit reduction RCH-equivalence 

摘      要：In this paper,we give the geometric constraint conditions of a canonical symplectic form and regular reduced symplectic forms for the dynamical vector fields of a regular controlled Hamiltonian(RCH)system and its regular reduced systems,which are called the Type I and Type II Hamilton-Jacobi equations.First,we prove two types of Hamilton-Jacobi theorems for an RCH system on the cotangent bundle of a configuration manifold by using the canonical symplectic form and its dynamical vector field.Second,we generalize the above results for a regular reducible RCH system with symmetry and a momentum map,and derive precisely two types of Hamilton-Jacobi equations for the regular point reduced RCH system and the regular orbit reduced RCH system.Third,we prove that the RCH-equivalence for the RCH system,and the RpCH-equivalence and RoCH-equivalence for the regular reducible RCH systems with symmetries,leave the solutions of corresponding Hamilton-Jacobi equations invariant.Finally,as an application of the theoretical results,we show the Type I and Type II Hamilton-Jacobi equations for the Rp-reduced controlled rigid body-rotor system and the Rp-reduced controlled heavy top-rotor system on the generalizations of the rotation group SO(3)and the Euclidean group SE(3),respectively.This work reveals the deeply internal relationships of the geometrical structures of phase spaces,the dynamical vector fields and the controls of the RCH system.