Hamilton-Jacobi Equations of Nonholonomic Magnetic Hamiltonian Systems
Abstract
In order to describe the impact of different geometric structures and constraints for the dynamics of a Hamiltonian system, in this paper, for a magnetic Hamiltonian system defined by a magnetic symplectic form, we first drive precisely the geometric constraint conditions of magnetic symplectic form for the magnetic Hamiltonian vector field.which are called the Type I and Type II of Hamilton-Jacobi equation. Secondly, for the magnetic Hamiltonian system with nonholonomic constraint, we first define a distributional magnetic Hamiltonian system, then derive its two types of Hamilton-Jacobi equation. 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 two types of Hamilton-Jacobi theorem. These research work reveal the deeply internal relationships of the magnetic symplectic structure, nonholonomic constraint, the distributional two-form, and the dynamical vector field of the nonholonomic magnetic Hamiltonian system.
- Publication:
-
arXiv e-prints
- Pub Date:
- December 2021
- DOI:
- 10.48550/arXiv.2112.00961
- arXiv:
- arXiv:2112.00961
- Bibcode:
- 2021arXiv211200961W
- Keywords:
-
- Mathematics - Symplectic Geometry;
- Mathematics - Differential Geometry;
- Mathematics - Dynamical Systems;
- 53D20;
- 70H20;
- 70F25
- E-Print:
- 27 pages. arXiv admin note: substantial text overlap with arXiv:2110.14175, arXiv:1508.07548