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Journal of Geometric Mechanics最新文献

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Local and global integrability of Lie brackets 李括号的局部和全局可积性
IF 0.8 4区 数学
Journal of Geometric Mechanics Pub Date : 2021-04-15 DOI: 10.3934/jgm.2021024
R. Fernandes, Yuxuan Zhang
{"title":"Local and global integrability of Lie brackets","authors":"R. Fernandes, Yuxuan Zhang","doi":"10.3934/jgm.2021024","DOIUrl":"https://doi.org/10.3934/jgm.2021024","url":null,"abstract":"<p style='text-indent:20px;'>We survey recent results on the local and global integrability of a Lie algebroid, as well as the integrability of infinitesimal multiplicative geometric structures on it.</p>","PeriodicalId":49161,"journal":{"name":"Journal of Geometric Mechanics","volume":"31 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86534201","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: 0
Quotients of double vector bundles and multigraded bundles 双向量束和多重梯度束的商
IF 0.8 4区 数学
Journal of Geometric Mechanics Pub Date : 2021-04-05 DOI: 10.3934/jgm.2021027
E. Meinrenken
{"title":"Quotients of double vector bundles and multigraded bundles","authors":"E. Meinrenken","doi":"10.3934/jgm.2021027","DOIUrl":"https://doi.org/10.3934/jgm.2021027","url":null,"abstract":"We study quotients of multi-graded bundles, including double vector bundles. Among other things, we show that any such quotient fits into a tower of affine bundles. Applications of the theory include a construction of normal bundles for weighted submanifolds, as well as for pairs of submanifolds with clean intersection.","PeriodicalId":49161,"journal":{"name":"Journal of Geometric Mechanics","volume":"2 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83500589","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: 0
Local convexity for second order differential equations on a Lie algebroid 李代数上二阶微分方程的局部凸性
IF 0.8 4区 数学
Journal of Geometric Mechanics Pub Date : 2021-03-26 DOI: 10.3934/jgm.2021021
J. Marrero, D. D. Diego, E. Mart'inez
{"title":"Local convexity for second order differential equations on a Lie algebroid","authors":"J. Marrero, D. D. Diego, E. Mart'inez","doi":"10.3934/jgm.2021021","DOIUrl":"https://doi.org/10.3934/jgm.2021021","url":null,"abstract":"<p style='text-indent:20px;'>A theory of local convexity for a second order differential equation (${text{sode}}$) on a Lie algebroid is developed. The particular case when the ${text{sode}}$ is homogeneous quadratic is extensively discussed.</p>","PeriodicalId":49161,"journal":{"name":"Journal of Geometric Mechanics","volume":"4 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86079128","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
Transitive double Lie algebroids via core diagrams 通过核图的传递双李代数
IF 0.8 4区 数学
Journal of Geometric Mechanics Pub Date : 2021-03-24 DOI: 10.3934/JGM.2021023
M. J. Lean, K. Mackenzie
{"title":"Transitive double Lie algebroids via core diagrams","authors":"M. J. Lean, K. Mackenzie","doi":"10.3934/JGM.2021023","DOIUrl":"https://doi.org/10.3934/JGM.2021023","url":null,"abstract":"The core diagram of a double Lie algebroid consists of the core of the double Lie algebroid, together with the two core-anchor maps to the sides of the double Lie algebroid. If these two core-anchors are surjective, then the double Lie algebroid and its core diagram are called transitive. This paper establishes an equivalence between transitive double Lie algebroids, and transitive core diagrams over a fixed base manifold. In other words, it proves that a transitive double Lie algebroid is completely determined by its core diagram.The comma double Lie algebroid associated to a morphism of Lie algebroids is defined. If the latter morphism is one of the core-anchors of a transitive core diagram, then the comma double algebroid can be quotiented out by the second core-anchor, yielding a transitive double Lie algebroid, which is the one that is equivalent to the transitive core diagram.Brown's and Mackenzie's equivalence of transitive core diagrams (of Lie groupoids) with transitive double Lie groupoids is then used in order to show that a transitive double Lie algebroid with integrable sides and core is automatically integrable to a transitive double Lie groupoid.","PeriodicalId":49161,"journal":{"name":"Journal of Geometric Mechanics","volume":"101 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78082082","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: 1
Holonomy transformations for Lie subalgebroids 李子代数群的完整变换
IF 0.8 4区 数学
Journal of Geometric Mechanics Pub Date : 2021-03-18 DOI: 10.3934/jgm.2021016
M. Zambon
{"title":"Holonomy transformations for Lie subalgebroids","authors":"M. Zambon","doi":"10.3934/jgm.2021016","DOIUrl":"https://doi.org/10.3934/jgm.2021016","url":null,"abstract":"Given a foliation, there is a well-known notion of holonomy, which can be understood as an action that differentiates to the Bott connection on the normal bundle. We present an analogous notion for Lie subalgebroids, consisting of an effective action of the minimal integration of the Lie subalgebroid, and provide an explicit description in terms of conjugation by bisections. The construction is done in such a way that it easily extends to singular subalgebroids, which provide our main motivation.","PeriodicalId":49161,"journal":{"name":"Journal of Geometric Mechanics","volume":"12 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84629483","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
The principle of virtual work and Hamilton's principle on Galilean manifolds 虚功原理和伽利略流形上的哈密顿原理
IF 0.8 4区 数学
Journal of Geometric Mechanics Pub Date : 2021-01-01 DOI: 10.3934/JGM.2021002
G. Capobianco, T. Winandy, S. Eugster
{"title":"The principle of virtual work and Hamilton's principle on Galilean manifolds","authors":"G. Capobianco, T. Winandy, S. Eugster","doi":"10.3934/JGM.2021002","DOIUrl":"https://doi.org/10.3934/JGM.2021002","url":null,"abstract":"To describe time-dependent finite-dimensional mechanical systems, their generalized space-time is modeled as a Galilean manifold. On this basis, we present a geometric mechanical theory that unifies Lagrangian and Hamiltonian mechanics. Moreover, a general definition of force is given, such that the theory is capable of treating nonpotential forces acting on a mechanical system. Within this theory, we elaborate the interconnections between classical equations known from analytical mechanics such as the principle of virtual work, Lagrange's equations of the second kind, Hamilton's equations, Lagrange's central equation, Hamel's generalized central equation as well as Hamilton's principle.","PeriodicalId":49161,"journal":{"name":"Journal of Geometric Mechanics","volume":"4 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84729896","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: 1
On nomalized differentials on spectral curves associated with the sinh-Gordon equation sinh-Gordon方程谱曲线上的归一化微分
IF 0.8 4区 数学
Journal of Geometric Mechanics Pub Date : 2021-01-01 DOI: 10.3934/jgm.2020023
T. Kappeler, Yannick Widmer
{"title":"On nomalized differentials on spectral curves associated with the sinh-Gordon equation","authors":"T. Kappeler, Yannick Widmer","doi":"10.3934/jgm.2020023","DOIUrl":"https://doi.org/10.3934/jgm.2020023","url":null,"abstract":"The spectral curve associated with the sinh-Gordon equation on the torus is defined in terms of the spectrum of the Lax operator appearing in the Lax pair formulation of the equation. If the spectrum is simple, it is an open Riemann surface of infinite genus. In this paper we construct normalized differentials on this curve and derive estimates for the location of their zeroes, needed for the construction of angle variables.","PeriodicalId":49161,"journal":{"name":"Journal of Geometric Mechanics","volume":"22 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91233066","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: 1
Book review: General theory of Lie groupoids and Lie algebroids, by Kirill C. H. Mackenzie 书评：Lie groupoids and Lie algebroids 的一般理论》，Kirill C. H. Mackenzie 著
IF 0.8 4区 数学
Journal of Geometric Mechanics Pub Date : 2021-01-01 DOI: 10.3934/jgm.2021026
T. Voronov
{"title":"Book review: General theory of Lie groupoids and Lie algebroids, by Kirill C. H. Mackenzie","authors":"T. Voronov","doi":"10.3934/jgm.2021026","DOIUrl":"https://doi.org/10.3934/jgm.2021026","url":null,"abstract":"<jats:p xml:lang=\"fr\" />","PeriodicalId":49161,"journal":{"name":"Journal of Geometric Mechanics","volume":"52 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86643585","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: 0
Quasi-bi-Hamiltonian structures and superintegrability: Study of a Kepler-related family of systems endowed with generalized Runge-Lenz integrals of motion 准双哈密顿结构与超可积性:具有广义朗格-伦兹运动积分的开普勒相关系统族的研究
IF 0.8 4区 数学
Journal of Geometric Mechanics Pub Date : 2021-01-01 DOI: 10.3934/JGM.2021003
M. F. Ranada
{"title":"Quasi-bi-Hamiltonian structures and superintegrability: Study of a Kepler-related family of systems endowed with generalized Runge-Lenz integrals of motion","authors":"M. F. Ranada","doi":"10.3934/JGM.2021003","DOIUrl":"https://doi.org/10.3934/JGM.2021003","url":null,"abstract":"The existence of quasi-bi-Hamiltonian structures for a two-dimen-sional superintegrable begin{document}$ (k_1,k_2,k_3) $end{document} -dependent Kepler-related problem is studied. We make use of an approach that is related with the existence of some complex functions which satisfy interesting Poisson bracket relations and that was previously applied to the standard Kepler problem as well as to some particular superintegrable systems as the Smorodinsky-Winternitz (SW) system, the Tremblay-Turbiner-Winternitz (TTW) and Post-Winternitz (PW) systems. We prove that these complex functions are important for two reasons: first, they determine the integrals of motion, and second they determine the existence of some geometric structures (in this particular case, quasi-bi-Hamiltonian structures). All the results depend on three parameters ( begin{document}$ k_1, k_2, k_3 $end{document} ) in such a way that in the particular case begin{document}$ k_1ne 0 $end{document} , begin{document}$ k_2 = k_3 = 0 $end{document} , the properties characterizing the Kepler problem are obtained. This paper can be considered as divided in two parts and every part presents a different approach (different complex functions and different quasi-bi-Hamil-tonian structures).","PeriodicalId":49161,"journal":{"name":"Journal of Geometric Mechanics","volume":"20 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80255466","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: 0
A Lagrangian approach to extremal curves on Stiefel manifolds Stiefel流形上极值曲线的拉格朗日方法
IF 0.8 4区 数学
Journal of Geometric Mechanics Pub Date : 2021-01-01 DOI: 10.3934/jgm.2020031
K. Hüper, I. Markina, F. Leite
{"title":"A Lagrangian approach to extremal curves on Stiefel manifolds","authors":"K. Hüper, I. Markina, F. Leite","doi":"10.3934/jgm.2020031","DOIUrl":"https://doi.org/10.3934/jgm.2020031","url":null,"abstract":"A unified framework for studying extremal curves on real Stiefel manifolds is presented. We start with a smooth one-parameter family of pseudo-Riemannian metrics on a product of orthogonal groups acting transitively on Stiefel manifolds. In the next step Euler-Langrange equations for a whole class of extremal curves on Stiefel manifolds are derived. This includes not only geodesics with respect to different Riemannian metrics, but so-called quasi-geodesics and smooth curves of constant geodesic curvature, as well. It is shown that they all can be written in closed form. Our results are put into perspective to recent related work where a Hamiltonian rather than a Lagrangian approach was used. For some specific values of the parameter we recover certain well-known results.","PeriodicalId":49161,"journal":{"name":"Journal of Geometric Mechanics","volume":"29 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87520744","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: 11
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