论文标题
非单学控制的哈密顿系统：对称减少和汉密尔顿 - 雅各比方程
Nonholonomic Controlled Hamiltonian System: Symmetric Reduction and Hamilton-Jacobi Equations
论文作者
论文摘要
为了在本文中描述非单位限制对常规受控的哈密顿量（RCH）系统动力学的影响，对具有非义学约束的RCH系统，我们首先通过仔细分析非独家RCH系统的动态矢量场的结构来得出其分布RCH系统。其次，我们精确地得出了分布RCH系统的动力学矢量场的诱导分布两种形式的几何约束条件，这称为汉密尔顿 - 雅各比方程的I型和II型II型。第三，我们将上述结果概括为具有对称性的非自我降低的RCH系统，并证明了两种类型的Hamilton-Jacobi定理，用于非世俗的降低分布RCH系统。此外，我们通过与常规点和规则轨道降低理论以及RCH系统动力学分析结合使用动量图的非独立降低RCH系统，我们给出了非霍马克常规点降低的几何形式，并且轨道降低了分布RCH系统，并证明了两种类型的Hamilton-JaCobi theorems用于这些分配系统。这些研究揭示了非物质约束，诱导的（分别减少）分布两种形式的深厚内部关系，非体力学RCH系统及其（降低）分布RCH系统的动力学矢量场和控制。
In order to describe the impact of nonholonomic constraints for the dynamics of a regular controlled Hamiltonian (RCH) system, in this paper, for an RCH system with nonholonomic constraint, we first derive its distributional RCH system, by analyzing carefully the structure of dynamical vector field of the nonholonomic RCH system. Secondly, we derive precisely the geometric constraint conditions of the induced distributional two-form for the dynamical vector field of the distributional RCH system, which are called the Type I and Type II of Hamilton-Jacobi equations. Thirdly, we generalize the above results for the nonholonomic reducible RCH system with symmetry, and prove two types of Hamilton-Jacobi theorems for the nonholonomic reduced distributional RCH system. Moreover, we consider the nonholonomic reducible RCH system with momentum map, by combining with the regular point and regular orbit reduction theory and the analysis of dynamics of RCH system, we give the geometric formulations of the nonholonomic regular point reduced and orbit reduced distributional RCH systems, and prove two types of Hamilton-Jacobi theorems for these reduced distributional RCH systems. These researches reveal the deeply internal relationships of the nonholonomic constraints, the induced (resp. reduced) distributional two-forms, the dynamical vector fields and controls of the nonholonomic RCH system and its (reduced) distributional RCH systems.
