Olivier Brahic
Motivated by the relationship between symplectic fibrations and classical Yang-Mills theories, we study the closeness of a $n$-form (n=2,3) defined on the total space of a fibration as a simple model for an abstract field theory. We introduce 2-plectic fibrations and interpret geometrically the corresponding equations for coupling in terms of higher analogues of connections.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
Symmetries and infinitesimal symmetries of singular differential equations
本文的目的是研究线性奇异微分方程的对称性，即由于导数与奇异线性算子相乘而不能写成正规形式的方程。引入线性奇异微分方程的几何对称性概念，作为保留定义问题的几何数据的变换。证明了这种对称性本质上等价于动态对称性，即解到解的变换。对于无穷小对称也给出了类似的结果。为了研究向量场流下几个对象的不变性，对无穷小变量进行了仔细的研究，特别强调无穷小向量丛自同构。
The phase space of relativistic particle mechanics is defined as the 1st jet space of motions regarded as timelike 1-dimensional submanifolds of spacetime. A Lorentzian metric and an electromagnetic 2-form define naturally on the odd-dimensional phase space a generalized contact structure. In the paper infinitesimal symmetries of the phase structures are characterized. More precisely, it is proved that all phase infinitesimal symmetries are special Hamiltonian lifts of distinguished conserved quantities on the phase space. It is proved that generators of infinitesimal symmetries constitute a L…
In this paper, we extend the well-known Noether theorem for Lagrangian systems to contact Lagrangian systems. We introduce a classification of infinitesimal symmetries and obtain the corresponding dissipated quantities. We notice that in contact dynamics, the existence of infinitesimal symmetries does not produce conserved quantities, but functions that dissipate at the same rate than the energy; so, the corresponding quotients are true conserved quantities.
We study non-flat planar 3-webs with infinitesimal symmetries. Using multi-dimensional Schwarzian derivative we give a criterion for linearization of such webs and present a projective classification thereof. Using this classification we show that the Gronwall conjecture is true for 3-webs admitting infinitesimal symmetries.
We study Lie algebras of generators of infinitesimal symmetries of almost-cosymplectic-contact structures of odd dimensional manifolds. The almost-cosymplectic-contact structure admits on the sheaf of pairs of 1-forms and functions the structure of a Lie algebra. We describe Lie subalgebras in this Lie algebra given by pairs generating infinitesimal symmetries of basic tensor fields given by the almost-cosymplectic-contact structure.
We introduce the infinitesimal symmetries of Dixmier-Douady gerbes over a manifold M, both with and without connective structures and curvings. We explore the algebraic structure possessed by these symmetries, and relate them to equivariant gerbes via a "differentiation functor". In the case that a gerbe G is equipped with a connective structure A, we give a new construction of the Courant algebroid associated to (G,A) directly in terms of the infinitesimal symmetries of (G,A).
The geometrical theory of partial differential equations in the absolute sense, without any additional structures, is developed. In particular the symmetries need not preserve the hierarchy of independent and dependent variables. The order of derivatives can be changed and the article is devoted to the higher--order infinitesimal symmetries which provide a simplifying "linear aproximation" of general groups of higher--order symmetries. The classical Lie's approach is appropriately adapted.
In this companion paper to our article {\em Accidental CR structures} (arxiv.org, January 2023), thought of as an appendix not submitted for publication, we provide complete explicit lists of infinitesimal CR automorphisms for the concerned CR models having respective Lie algebra structures: $${\bf E}_{II}, \qquad\ {\bf E}_{III}, \qquad\ \mathfrak{so}(\ell-1,\ell+1), \qquad\ \mathfrak{su}(p,q).$$ We start from our lists of {\em quadric} CR submanifolds $M^{2n+c} \subset \mathbb{C}^{n+c}$ of codimension $c >1$ which are shown to be {\em accidental}, in the sense that their CR symmetry groups ar…
Using the procedure initiated in \cite{Ma2013}, we deform Lax-type equations though a scaling of the time parameter. This gives an equivalent (deformed) equation which is integrable in terms of power series of the scaling parameter. We then describe a regular Frölicher Lie group of symmetries of this deformed equation
In the complex setting, let $F(x,y,y')=0$ be an analytic or algebraic differential equation with $y'$-degree $d$. We deal with the qualitative study of such equations through the geometry of the planar $d$-web generated by the generic family of integral curves. Infinitesimal symmetries of these configurations are discussed by using meromorphic connections methods "à la Cartan-Spencer".
We provide a complete classification of hexagonal singular 3-web germs in the complex plane, satisfying the following two conditions: 1) the Chern connection remains holomorphic at the singular point, 2) the web admits at least one infinitesimal symmetry at this point. As a by-product, a classification of hexagonal weighted homogeneous 3-webs is obtained.
In this paper we study the infinitesimal symmetries, Newtonoid vector fields, infinitesimal Noether symmetries and conservation laws of Hamiltonian systems. Using the dynamical covariant derivative and Jacobi endomorphism on the cotangent bundle we find the invariant equations of infinitesimal symmetries and Newtonoid vector fields and prove that the canonical nonlinear connection induced by a regular Hamiltonian can be determined by these symmetries. Finally, an example from optimal control theory is given.
Geometric structures as deformed infinitesimal symmetries
通过变形无穷小对称，得到了微分流形上几何结构的一般模型。具体来说，该模型由一个李代数体，配备了一个仿射联络兼容的李代数体结构。当结构局部对称时，这种连接的曲率恰好为零。该模型将Cartan几何这一重要的几何类推广到了不传递的情形。简单的例子进行调查，并确定相应的局部障碍对称。这些例子包括叶理、黎曼结构、无穷小G结构、辛结构和泊松结构。
Infinitesimal symmetries of bundle gerbes and Courant algebroids
本文证明了光滑流形上 χ-扭曲 Courant 代数胚截面的 Lie 2-代数与 3-曲率为 χ 的 S^1-束 gerbe 上保联络乘法向量场的 Lie 2-代数拟同构，建立了二者的无穷小对称性对应。
On second Noether's theorem and gauge symmetries in Mechanics
我们回顾了含时力学中第二Noether定理的几何表述。研究了最终约束流形上的动力学与对称的无穷小生成元之间的对易关系。我们展示了一个算法，用于确定规范的对称性，这是密切相关的过程中的稳定的约束，无论是在拉格朗日和哈密顿形式主义。这两种形式主义之间的连接建立的时间演化算子。
Symmetries of Differential Equations via Cartan's Method of Equivalence
基于Cartan的等价方法和Fels和Olver提出的移动上标架方法，给出了计算微分方程对称伪群的不变1-形式和结构方程的方法.该方法不需要对无穷小定义系统进行初步的计算、分析和积分，只使用微分和线性代数运算。给出了应用实例。
More On Generalized Symmetries of Partial Differential Equations and Quasiexact Solvability
对于一类系统的偏微分方程，无穷小的平移（关于一些（在）因变量）具有特定的有限维不变子空间的广义对称的空间考虑系统。我们建立了当这些子空间存在广义对称时，它们依赖于上述变量。
Symmetries of Contact Metric Manifolds
研究了紧致Sasakian流形和K-切触流形上的无穷小等距的李代数。在非空间型或3-Sasaki流形上，每个Killing向量场都是Sasaki结构的无穷小自同构。对具有K-切触结构的流形，证明了存在一个非该结构的无穷小自同构的常长Killing向量场当且仅当该流形是由紧致伪黎曼四元数Kaehler流形的Konishi丛在一个极大负分布上变换度量的符号后得到的.证明了非正则Sasakian流形不是齐次流形，并构造了上齐次流形的例子。利用这些结果，我们在最后一节中得到了所有齐次Sasakian流形的分类。
Local Integrable Symmetries of Diffieties
本文研究了diffieties框架下的局部可积对称性，通过求解偏微分方程系统来分析diffiety的局部分类及平坦性测试。
The results of this paper have been greatly superseded by those in the paper "Contact geometry and isosystolic inequalities" (arXiv:1109.4253) by the same authors.
On symmetries in Galilei classical mechanics
在伽利略经典力学的框架中（即，在Jadczyk和Modugno发展的时空上的广义相对论性经典力学的基础上，系统地分析了几何对象的对称性之间的关系。我们表明，（完整的）无穷小对称性的共辛结构的时空和它的潜力也是对称性的类空度规，引力和电磁场，欧拉-拉格朗日态射，拉格朗日。然后，我们利用相空间函数的协变提升，给出了一个与一组共辛对称相关联的协变动量映射。在投射到时空上的作用量的情况下，我们看到，这个动量图的分量是在Jadczyck和Modugno意义下的可量化函数。最后，我们用一些例子说明了结果。
A model-free method for discovering symmetry in differential equations
针对无控制方程显式知识时从离散数据识别李对称性的挑战，提出基于广义移动最小二乘法延拓数据并构建线性系统的无模型方法，推导收敛界，实验验证其准确性与鲁棒性，助力动力系统对称性的发现。
Generalized Symmetries of Partial Differential Equations and Quasiexact Solvability
利用无穷小平移（关于某些（不）相依变量）在某些偏微分方程组的广义对称空间的特定有限维子空间上的伴随作用，我们明确地确定了这些子空间的广义对称系数对上述变量的相依性.我们建立了我们的结果与理论的准精确可解模型的连接。还讨论了所提出的方法的一些推广。
This paper is concerned with symmetries of closed multiplicative 2-forms on Lie groupoids and their infinitesimal counterparts. We use them to study Lie group actions on Dirac manifolds by Dirac diffeomorphisms and their lifts to presymplectic groupoids, building on recent work of Fernandes-Ortega-Ratiu \cite{FOR} on Poisson actions.
We study the interplay between the differential Galois group and the Lie algebra of infinitesimal symmetries of systems of linear differential equations. We show that some symmetries can be seen as solutions of a hierarchy of linear differential systems. We show that the existence of rational symmetries constrains the differential Galois group in the system in a way that depends of the Maclaurin series of the symmetry along the zero solution.
We present a new link between the Invariant Theory of infinitesimal singular Riemannian foliations and Jordan algebras. This, together with an inhomogeneous version of Weyl's First Fundamental Theorems, provides a characterization of the recently discovered Clifford foliations in terms of basic polynomials. This link also yields new structural results about infinitesimal foliations, such as the existence of non-trivial symmetries.
Lie algebroid morphisms, Poisson Sigma Models, and off-shell closed gauge symmetries
三维的陈-西蒙斯规范理论和二维的泊松-西格玛模型（PSM）是同一理论的例子，如果它们的场方程被解释为李代数胚的态射，它们的对称性（在壳上）被解释为这种态射的同伦。我们指出，（离壳）规范对称性的PSM在文献中不是全球良好定义的非平行泊松流形，并提出了一个协变定义，他们作为左作用的一些有限维李代数体。我们的方法可以避免BV-和AKSZ-形式主义的无限维超几何中出现的并发症。这个预印本是一系列论文的起点，旨在介绍李代数胚的杨-米尔斯型规范理论，其中包括和推广标准YM理论，PSM和gerbes。
Let $Q$ be a closed manifold admitting a locally-free action of a compact Lie group $G$. In this paper we study the properties of geodesic flows on $Q$ given by Riemannian metrics which are invariant by such an action. In particular, we will be interested in the existence of geodesics which are closed up to the action of some element in the group $G$, since they project to closed magnetic geodesics on the quotient orbifold $Q/G$.
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。