Hamilton-Jacobi Equations of Controlled Magnetic Hamiltonian Systems

In this paper, we first define a kind of controlled magnetic Hamiltonian (CMH) system, and give a good expression of the dynamical vector field of the CMH system, such that we can describe the magnetic vanishing condition and the CMH-equivalence, and derive precisely the geometric constraint conditions of the magnetic symplectic form for the dynamical vector field of the CMH system, which are called the Type I and Type II of Hamilton-Jacobi equation. Next, we prove that the CMH-equivalence for the CMH systems leaves the solutions of the associated Hamilton-Jacobi equations invariant, if the associated magnetic Hamiltonian systems are equivalent. These research works reveal the deeply internal relationships of the magnetic symplectic forms, the dynamical vector fields and controls of the CMH systems.