(untitled) (Communications in Mathematical Research, 2022)
In order to describe the impact of the different geometric structures and the constraints for the dynamics of a Hamiltonian system, in this paper, for a magnetic Hamiltonian system defined by a magnetic symplectic form, we drive precisely the geometric constraint conditions of the magnetic symplectic form for the magnetic Hamiltonian vector field, which are called the Type I and Type II Hamilton-Jacobi equations. Second, for the magnetic Hamiltonian system with a nonholonomic constraint, we can define a distributional magnetic Hamiltonian system, then derive its two types of Hamilton-Jacobi equations. Moreover, we generalize the above results to nonholonomic reducible magnetic Hamiltonian system with symmetry, we define a nonholonomic reduced distributional magnetic Hamiltonian system, and prove the two types of Hamilton-Jacobi theorems. These research reveal the deeply internal relationships of the magnetic symplectic structure, the nonholonomic constraint, the distributional two-form, and the dynamical vector field of the nonholonomic magnetic Hamiltonian system.