Journal of Geometric Mechanics最新文献_第8页

Relative periodic solutions of the begin{document}$ n $end{document}-vortex problem on the sphere Relative periodic solutions of the begin{document}$ n $end{document}-vortex problem on the sphere

IF 0.8 4区数学

Journal of Geometric Mechanics Pub Date : 2019-08-21 DOI: 10.3934/JGM.2019021

C. García-Azpeitia

引用次数: 2

Generalized point vortex dynamics on begin{document}$ mathbb{CP} ^2 $end{document} Generalized point vortex dynamics on begin{document}$ mathbb{CP} ^2 $end{document}

IF 0.8 4区数学

Journal of Geometric Mechanics Pub Date : 2019-07-17 DOI: 10.3934/jgm.2019030

J. Montaldi, Amna Shaddad

{"title":"Generalized point vortex dynamics on begin{document}$ mathbb{CP} ^2 $end{document}","authors":"J. Montaldi, Amna Shaddad","doi":"10.3934/jgm.2019030","DOIUrl":"https://doi.org/10.3934/jgm.2019030","url":null,"abstract":"This is the second of two companion papers. We describe a generalization of the point vortex system on surfaces to a Hamiltonian dynamical system consisting of two or three points on complex projective space begin{document}$ mathbb{CP} ^2 $end{document} interacting via a Hamiltonian function depending only on the distance between the points. The system has symmetry group SU(3). The first paper describes all possible momentum values for such systems, and here we apply methods of symplectic reduction and geometric mechanics to analyze the possible relative equilibria of such interacting generalized vortices. The different types of polytope depend on the values of the 'vortex strengths', which are manifested as coefficients of the symplectic forms on the copies of begin{document}$ mathbb{CP} ^2 $end{document} . We show that the reduced space for this Hamiltonian action for 3 vortices is generically a 2-sphere, and proceed to describe the reduced dynamics under simple hypotheses on the type of Hamiltonian interaction. The other non-trivial reduced spaces are topological spheres with isolated singular points. For 2 generalized vortices, the reduced spaces are just points, and the motion is governed by a collective Hamiltonian, whereas for 3 the reduced spaces are of dimension at most 2. In both cases the system will be completely integrable in the non-abelian sense.","PeriodicalId":49161,"journal":{"name":"Journal of Geometric Mechanics","volume":"19 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2019-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73263679","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 4

Non-Abelian momentum polytopes for products of begin{document}$ mathbb{CP}^2 $end{document} Non-Abelian momentum polytopes for products of begin{document}$ mathbb{CP}^2 $end{document}

IF 0.8 4区数学

Journal of Geometric Mechanics Pub Date : 2019-06-17 DOI: 10.3934/jgm.2019029

J. Montaldi, Amna Shaddad

引用次数: 1

Dual pairs for matrix groups 矩阵群的对偶对

IF 0.8 4区数学

Journal of Geometric Mechanics Pub Date : 2019-05-06 DOI: 10.3934/JGM.2019014

Paul Skerritt, Cornelia Vizman

引用次数: 5

Riemannian cubics and elastica in the manifold begin{document}$ operatorname{SPD}(n) $end{document} of all begin{document}$ ntimes n $end{document} symmetric positive-definite matrices Riemannian cubics and elastica in the manifold begin{document}$ operatorname{SPD}(n) $end{document} of all begin{document}$ ntimes n $end{document} symmetric positive-definite matrices

IF 0.8 4区数学

Journal of Geometric Mechanics Pub Date : 2019-05-06 DOI: 10.3934/JGM.2019015

Erchuan Zhang, L. Noakes

{"title":"Riemannian cubics and elastica in the manifold begin{document}$ operatorname{SPD}(n) $end{document} of all begin{document}$ ntimes n $end{document} symmetric positive-definite matrices","authors":"Erchuan Zhang, L. Noakes","doi":"10.3934/JGM.2019015","DOIUrl":"https://doi.org/10.3934/JGM.2019015","url":null,"abstract":"Left Lie reduction is a technique used in the study of curves in bi-invariant Lie groups [ 32 , 33 , 40 ]. Although the manifold begin{document}$ operatorname{SPD}(n) $end{document} of all begin{document}$ ntimes n $end{document} symmetric positive-definite matrices is not a Lie group with respect to the standard matrix multiplication, it is a symmetric space with a left action of begin{document}$ GL(n) $end{document} and an isotropy group begin{document}$ SO(n) $end{document} leaving the identity matrix fixed. The main purpose of this paper is to extend the method of left Lie reduction to begin{document}$ operatorname{SPD}(n) $end{document} and use it to study two second order variational curves: Riemannian cubics and elastica. Riemannian cubics in begin{document}$ operatorname{SPD}(n) $end{document} are reduced to so-called Lie quadratics in the Lie algebra begin{document}$ mathfrak{gl}(n) $end{document} and geometric analyses are presented. Besides, by using the Frenet-Serret frames and the extended left Lie reduction separately, we investigate elastica in the manifold begin{document}$ operatorname{SPD}(n) $end{document} . The latter presents a comparatively simple form of the equations for elastica in begin{document}$ operatorname{SPD}(n) $end{document} .","PeriodicalId":49161,"journal":{"name":"Journal of Geometric Mechanics","volume":"83 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2019-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91393647","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 2

Infinite dimensional integrals and partial differential equations for stochastic and quantum phenomena 随机和量子现象的无限维积分和偏微分方程

IF 0.8 4区数学

Journal of Geometric Mechanics Pub Date : 2019-05-06 DOI: 10.3934/JGM.2019006

S. Albeverio, S. Mazzucchi

引用次数: 0

Riemann-Hilbert problem, integrability and reductions 黎曼-希尔伯特问题，可积性和约简

IF 0.8 4区数学

Journal of Geometric Mechanics Pub Date : 2019-02-27 DOI: 10.3934/jgm.2019009

V. Gerdjikov, R. Ivanov, A. Stefanov

引用次数: 11

De Donder form for second order gravity 二阶重力的De Donder形式