|
|
Pure Mathematics 2024
圆环中2-点涡的可积运动
|
Abstract:
本文运用镜像法得到了圆环区域上 Dirichlet 边值拉普拉斯算子的格林函数和两点涡系统的哈密顿函数，并用作用-角变量方法对系统进行约化。 以两点涡强度相等为例，对圆环分别在内外半径比q = 0.02, q = 0.08, q = 0.2 时得到了以涡度矩 I 为参数的系统相对均衡解的完整分类，最后针对 各种情况刻画出两个点涡的相对运动轨迹。
In this paper, the Green's function of the Dirichlet marginal Laplace operator and the Hamiltonian function of the two-point vortex system in annular domain are obtained
by the method of image, we make reduction to the system by action-angle variables
method, and take the example of the equal strength case to describe their relative
motion trajectories and classify the system's quasi-equilibrium solutions were obtained
by considering the circular ring at different inner-to-outer radius ratios:q = 0:02,
q = 0:08 and q = 0:2, with the vorticity moment I as a parameter.
| [1] | Lin, C.C. (1941) On the Motion of Vortices in 2D: I. Existence of the Kirchhoff-
Routh Function. Proceedings of the National Academy of Sciences of the United
States of America, 27, 570-575. https://doi.org/10.1073/pnas.27.12.570 |
| [2] | Lin, C.C. (1941) On the Motion of Vortices in 2D II. Some Further Properties on
the Kirchhoff-Routh Function. Proceedings of the National Academy of Sciences of
the United States of America, 27, 575-577. https://doi.org/10.1073/pnas.27.12.575 |
| [3] | Thomson, J.J.S. (1968) A Treatise on the Motion of Vortex Rings: An Essay to
Which the Adams Prize Was Adjudged in 1882, in the University of Cambridge.
Dawsons of Pall Mall, London. |
| [4] | Havelock T. H. (1931) The Stability of Motion of Rectilinear Vortices in Ring For-
mation. Philosophical Magazine, 11, 617-633.
https://doi.org/10.1080/14786443109461714 |
| [5] | Kurakin, L.G. (2010) On the Stability of Thomson's Vortex Configurations inside a
Circular Domain. Regular and Chaotic Dynamics, 15, 40-58.
https://doi.org/10.1134/S1560354710010028 |
| [6] | Pashaev, O.K. and Yilmaz, O. (2011) Hamiltonian Dynamics of N Vortices in Con-
centric Annular Region. Journal of Physics A: Mathematical and Theoretical, 44,
Article 185501. https://doi.org/10.1088/1751-8113/44/18/185501 |
| [7] | Lakaniemi, M. (2007) On the Dynamics of Point Vortices in a Quantum Gas Con-
fined in an Annular Region. arXiv: 0708.1898 |
| [8] | Bolsinov, A.V., et al. (2010) Topology and Stability of Integrable Systems. Russian
Mathematical Surveys, 65, 259-318.
https://doi.org/10.1070/RM2010v065n02ABEH004672 |
| [9] | Vaskin, V.V. and Erdakova, N.N. (2010) On the Dynamics of Two Point Vortices in
an Annular Region. Russian Journal of Nonlinear Dynamics, 6, 531-547.
https://doi.org/10.20537/nd1003005 |
| [10] | Courant, R. and Hilbert, D. (1989) Methods of Mathematical physics. Vol. II. Partial
Differential Equations. John Wiley & Sons, Inc., New York.
https://doi.org/10.1002/9783527617234 |
| [11] | Arnold, V.I. (1978) Mathematical Methods of Classical Mechanics. Springer-Verlag,
New York. https://doi.org/10.1007/978-1-4757-1693-1 |
| [12] | 戴芋慧, 晋榕榕, 毛玉兰. 圆盘中2-点涡的可积运动[J]. 北京师范大学学报(自然科学
版), 2019, 55(2): 179-184. |
