arrow
Volume 38, Issue 3
Hamilton-Jacobi Equations for Nonholonomic Magnetic Hamiltonian Systems

Hong Wang

Commun. Math. Res., 38 (2022), pp. 351-388.

Published online: 2022-08

Export citation
  • Abstract

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.

  • AMS Subject Headings

70H20, 70F25, 53D20

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address
  • BibTex
  • RIS
  • TXT
@Article{CMR-38-351, author = {Wang , Hong}, title = {Hamilton-Jacobi Equations for Nonholonomic Magnetic Hamiltonian Systems}, journal = {Communications in Mathematical Research }, year = {2022}, volume = {38}, number = {3}, pages = {351--388}, abstract = {

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.

}, issn = {2707-8523}, doi = {https://doi.org/10.4208/cmr.2022-0028}, url = {http://global-sci.org/intro/article_detail/cmr/20961.html} }
TY - JOUR T1 - Hamilton-Jacobi Equations for Nonholonomic Magnetic Hamiltonian Systems AU - Wang , Hong JO - Communications in Mathematical Research VL - 3 SP - 351 EP - 388 PY - 2022 DA - 2022/08 SN - 38 DO - http://doi.org/10.4208/cmr.2022-0028 UR - https://global-sci.org/intro/article_detail/cmr/20961.html KW - Hamilton-Jacobi equation, magnetic Hamiltonian system, nonholonomic constraint, distributional magnetic Hamiltonian system, nonholonomic reduction. AB -

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.

Hong Wang. (2022). Hamilton-Jacobi Equations for Nonholonomic Magnetic Hamiltonian Systems. Communications in Mathematical Research . 38 (3). 351-388. doi:10.4208/cmr.2022-0028
Copy to clipboard
The citation has been copied to your clipboard