Hamilton-Jacobi Equations for Nonholonomic Magnetic Hamiltonian Systems (Communications in Mathematical Research, 2022)
In order to describe the impact of the different geometric structuresand the constraints for the dynamics of a Hamiltonian system, in this paper,for a magnetic Hamiltonian system defined by a magnetic symplectic form, wedrive precisely the geometric constraint conditions of the magnetic symplectic form for the magnetic Hamiltonian vector field, which are called the Type Iand Type II Hamilton-Jacobi equations. Second, for the magnetic Hamiltoniansystem 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 reduciblemagnetic Hamiltonian system with symmetry, we define a nonholonomic reduced distributional magnetic Hamiltonian system, and prove the two types ofHamilton-Jacobi theorems. These research reveal the deeply internal relationships of the magnetic symplectic structure, the nonholonomic constraint, thedistributional two-form, and the dynamical vector field of the nonholonomicmagnetic Hamiltonian system.