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 $R_p$-reduced controlled rigid body-rotor system and the $R_p$-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.
Hong Wang
. HAMILTON-JACOBI EQUATIONS FOR A REGULAR CONTROLLED HAMILTONIAN SYSTEM AND ITS REDUCED SYSTEMS*[J]. Acta mathematica scientia, Series B, 2023
, 43(2)
: 855
-906
.
DOI: 10.1007/s10473-023-0221-5
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