Joris Vankerschaver, Frans Cantrijn, Manuel de Leon, David Martin de Diego
A geometric model for nonholonomic Lagrangian field theory is studied. The multisymplectic approach to such a theory as well as the corresponding Cauchy formalism are discussed. It is shown that in both formulations, the relevant equations for the constrained system can be recovered by a suitable projection of the equations for the underlying free (i.e. unconstrained) Lagrangian system.
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A $k$-symplectic framework for classical field theories subject to nonholonomic constraints is presented. If the constrained problem is regular one can construct a projection operator such that the solutions of the constrained problem are obtained by projecting the solutions of the free problem. Symmetries for the nonholonomic system are introduced and we show that for every such symmetry, there exist a nonholonomic momentum equation. The proposed formalism permits to introduce in a simple way many tools of nonholonomic mechanics to nonholonomic field theories.
In this note, we introduce a suitable generalization of the momentum map for nonholonomic field theories and prove a covariant form of the nonholonomic momentum equation. We show that these covariant objects coincide with their counterparts in mechanics by making the transition to the Cauchy formalism.
The developments in this paper are concerned with nonholonomic field theories in the presence of symmetries. Having previously treated the case of vertical symmetries, we now deal with the case where the symmetry action can also have a horizontal component. As a first step in this direction, we derive a new and convenient form of the field equations of a nonholonomic field theory. Nonholonomic symmetries are then introduced as symmetry generators whose virtual work is zero along the constraint submanifold, and we show that for every such symmetry, there exists a so-called momentum equation, de…
A multisymplectic setting for classical field theories subjected to non-holonomic constraints is presented. The infinite dimensional setting in the space of Cauchy data is also given.
This paper has been inspired by ideas presented by V. V. Kozlov in his works [19, 20]. In this paper our goal is to carry out a thorough analysis of some geometric problems of the dynamics of affinely-rigid bodies. We present two ways to describe this case: the classical dynamical d'Alembert and variational, i.e., vakonomic one. So far, we can see that they give quite different results. The vakonomic model from the mathematical point of view seems to be more elegant. The similar problems were examined by Jòźwikowski and W. Respondek in their paper [16]
We give a geometric description of variational principles in mechanics, with special attention to constrained systems. For the general case of nonholonomic constraints, a unified variational approach is given, and the equations of motion of both vakonomic and nonholonomic frameworks are obtained. We study specifically the existence of infinitesimal variations in both cases. When the constraints are integrable, both formalisms are compared and it is proved that they coincide. As examples, we give geometric formulations of the equations of motion for the case of optimal control and for vakonomic…
We study stationary configurations mimicking nonholonomic locally anisotropic black rings (for instance, with ellipsoidal polarizations and/or imbedded into solitonic backgrounds) in three/six dimensional pseudo-Finsler/ Riemannian spacetimes. In the asymptotically flat limit, for holonomic configurations, a subclass of such spacetimes contains the set of five dimensional black ring solutions with regular rotating event horizon. For corresponding parameterizations, the metrics and connections define Finsler-Einstein geometries modeled on tangent bundles, or on nonholonomic (pseudo) Riemannian …
We study how massive ghost-free gravity $f(R)$-modified theories, MGFTs, can be encoded into generic off-diagonal Einstein spaces. Using "auxiliary" connections completely defined by the metric fields and adapted to nonholonomic frames with associated nonlinear connection structure, we decouple and integrate in certain general forms the field equations in MGFT. Imposing additional nonholonomic constraints, we can generate Levi-Civita, LC, configurations and mimic MGFT effects via off-diagonal interactions of effective Einstein and/or Einstein-Cartan gravity with nonholonomically induced torsio…
We elaborate an unified geometric approach to classical mechanics, Riemann-Finsler spaces and gravity theories on Lie algebroids provided with nonlinear connection (N-connection) structure. There are investigated the conditions when the fundamental geometric objects like the anchor, metric and linear connection, almost sympletic and related almost complex structures may be canonically defined by a N-connection induced from a regular Lagrangian (or Hamiltonian), in mechanical models, or by generic off-diagonal metric terms and nonholonomic frames, in gravity theories. Such geometric constructio…
We argue that the Einstein gravity theory can be reformulated in almost Kahler (nonsymmetric) variables with effective symplectic form and compatible linear connection uniquely defined by a (pseudo) Riemannian metric. A class of nonsymmetric theories of gravitation (NGT) on manifolds enabled with nonholonomic distributions is analyzed. There are considered some conditions when the fundamental geometric and physical objects are determined/ modified by nonholonomic deformations in general relativity or by contributions from Ricci flow theory and/or quantum gravity. We prove that in such NGT, for…
In this summary of Habilitation Thesis, it is outlined author's 18 years research activity on mathematical physics, geometric methods in particle physics and gravity, modifications and applications (after defending his PhD thesis in 1994). Ten most relevant publications are structured conventionally into three "strategic directions": 1) nonholonomic geometric flows evolutions and exact solutions for Ricci solitons and field equations in (modified) gravity theories; 2) geometric methods in quantization of models with nonlinear dynamics and anisotropic field interactions; 3) (non) commutative ge…
We study fractional configurations in gravity theories and Lagrange mechanics. The approach is based on Caputo fractional derivative which gives zero for actions on constants. We elaborate fractional geometric models of physical interactions and we formulate a method of nonholonomic deformations to other types of fractional derivatives. The main result of this paper consists in a proof that for corresponding classes of nonholonomic distributions a large class of physical theories are modelled as nonholonomic manifolds with constant matrix curvature. This allows us to encode the fractional dyna…
We briefly review our recent results on the geometry of nonholonomic manifolds and Lagrange--Finsler spaces and fractional calculus with Caputo derivatives. Such constructions are used for elaborating analogous models of fractional gravity and fractional Lagrange mechanics.
We study possible links between quantum gravity phenomenology encoding Lorentz violations as nonlinear dispersions, the Einstein-Finsler gravity models, EFG, and nonholonomic (non-integrable) deformations to Hořava-Lifshitz, HL, and/or Einstein's general relativity, GR, theories. EFG and its scaling anisotropic versions formulated as Hořava-Finsler models, HF, are constructed as covariant metric compatible theories on (co) tangent bundle to Lorentz manifolds and respective anisotropic deformations. Such theories are integrable in general form and can be quantized following standard methods of …
We use an important decoupling property of gravitational field equations in the general relativity theory and modifications, written with respect to nonholonomic frames with 2+2 spacetime decomposition. This allows us to integrate the Einstein equations (eqs) in very general forms with generic off--diagonal metrics depending on all spacetime coordinates via generating and integration functions containing (un--)broken symmetry parameters. We associate families of off-diagonal Einstein manifolds to certain classes of covariant gravity theories which have a nice ultraviolet behavior and seem to b…
We study geometric relativistic flow and Ricci soliton equations which (for respective nonholonomic constraints and self-similarity conditions) are equivalent to the gravitational field equations of $R^2$ gravity and/or to the Einstein equations with scalar field in general relativity, GR. Perelman's functionals are generalized for modified gravity theories, MGTs, which allows to formulate an analogous statistical thermodynamics for geometric flows and Ricci solitons. There are constructed and analyzed generic off-diagonal black ellipsoid, black hole and solitonic exact solutions in MGTs and G…
In this article, we review the current status of Finsler-Lagrange geometry and generalizations. The goal is to aid non-experts on Finsler spaces, but physicists and geometers skilled in general relativity and particle theories, to understand the crucial importance of such geometric methods for applications in modern physics. We also would like to orient mathematicians working in generalized Finsler and Kahler geometry and geometric mechanics how they could perform their results in order to be accepted by the community ''orthodox'' physicists. Although the bulk of former models of Finsler-Lagra…
Models of geometric flows pertaining to $\mathcal{R}^2$ scale invariant (super) gravity theories coupled to conformally invariant matter fields are investigated. Related to this work are supersymmetric scalar manifolds that are isomorphic to the Kählerian spaces $\mathcal{M}_n=SU(1,1+k)/U(1)\times SU(1+k)$ as generalizations of the non-supersymmetric analogs with $SO(1,1+k)/SO(1+k)$ manifolds. For curved superspaces with geometric evolution of physical objects, a complete supersymmetric theory has to be elaborated on nonholonomic (super) manifolds and bundles determined by non-integrable super…
The purpose of this paper is to define the concept of multi-Dirac structures and to describe their role in the description of classical field theories. We begin by outlining a variational principle for field theories, referred to as the Hamilton-Pontryagin principle, and we show that the resulting field equations are the Euler-Lagrange equations in implicit form. Secondly, we introduce multi-Dirac structures as a graded analog of standard Dirac structures, and we show that the graph of a multisymplectic form determines a multi-Dirac structure. We then discuss the role of multi-Dirac structures…
We investigate bi-Hamiltonian structures and mKdV hierarchies of solitonic equations generated by (semi) Riemannian metrics and curve flows of non-stretching curves. There are applied methods of the geometry of nonholonomic manifolds enabled with metric-induced nonlinear connection (N-connection) structure. On spacetime manifolds, we consider a nonholonomic splitting of dimensions and define a new class of liner connections which are 'N-adapted', metric compatible and uniquely defined by the metric structure. We prove that for such a linear connection, one yields couples of generalized sine-Go…
To appear in Encyclopedia of Mathematical Physics, published by Elsevier in early 2006. Comments/corrections welcome. The article surveys topological aspects in gauge theories.
General off-diagonal integrability of metric and nonmetric geometric flow and Finsler-Lagrange-Hamilton modified Einstein equations
本文探讨了度量和非度量几何流及修正爱因斯坦方程的非对角可积性，总结了新成果和历史发展，提出了一种非平凡的爱因斯坦引力修改方法。
Using double 2+2 and 3+1 nonholonomic fibrations on Lorentz manifolds, we extend the concept of W-entropy for gravitational fields in the general relativity, GR, theory. Such F- and W-functionals were introduced in the Ricci flow theory of three dimensional, 3-d, Riemannian metrics by G. Perelman, arXiv: math.DG/0211159. Nonrelativistic 3-d Ricci flows are characterized by associated statistical thermodynamical values determined by W--entropy. Generalizations for geometric flows of 4-d pseudo-Riemannian metrics are considered for models with local thermodynamical equilibrium and separation of …
On some aspects of the discretization of the Suslov problem
本文探讨了在SO(3)上Euler-Poincaré-Suslov方程的离散化，分析了不约简与约简设置下离散重构方程由Cayley重映射给出时的一致性阶关系，并给出了保证变分积分器保持约束分布的条件，同时通过图表展示了不同离散化方法的性能。
Bethe/Gauge correspodence: a short review on an aspect of the integrability nature of supersymmetric gauge theories
本文综述了 Bethe/Gauge 对应，以 XXX 自旋链与二维 N=(2,2) 规范理论为例说明基本思想，并通过比较瞬子与涡旋配分函数探讨四维与二维理论的可积性联系。
If a scalar field theory in (1+1) dimensions possesses soliton solutions obeying first order BPS equations, then, in general, it is possible to find an infinite number of related field theories with BPS solitons which obey closely related BPS equations. We point out that this fact may be understood as a simple consequence of an appropriately generalised notion of self-duality. We show that this self-duality framework enables us to generalize to higher dimensions the construction of new solitons from already known solutions. By performing simple field transformations our procedure allows us to …
Asymptotic safe nonassociative quantum gravity with star R-flux products, Goroff-Sagnotti counter-terms, and geometric flows
本研究针对非结合修正引力中渐近安全性受微扰项破坏的质疑，提出含星R-流乘积的非结合量子引力方案，通过非完整函数重整化技术证明其经典与量子层面均有良好定义，计算了相应的广义热力学变量。
Quantization of topological-holomorphic field theories: local aspects
本文在模型流形R^m×C^n上严格构造拓扑-全纯混合场论的微扰单圈量子化，证明当m≥1时单圈反常障碍消失，并由此得到纯N=2四维超对称Yang-Mills理论扭曲在单圈水平上存在精确有限量子化。
Real Hamiltonian forms of affine Toda field theories: spectral aspects
本文研究与例外单李代数相关的 2 维 Toda 场论的真实哈密顿形式及其 Lax 算子谱理论，通过阐述 Riemann-Hilbert 问题推导出唯一确定散射矩阵与算子势的最小散射数据集。
The variational formalism for classical field theories is extended to the setting of Lie algebroids. Given a Lagrangian function we study the problem of finding critical points of the action functional when we restrict the fields to be morphisms of Lie algebroids. In addition to the standard case, our formalism includes as particular examples the case of systems with symmetry (covariant Euler-Poincare and Lagrange Poincare cases), Sigma models or Chern-Simons theories.
This series of lectures is planned as a generalization of author's large (more than fifteen years) experience of work in the theoretical physics. The modern theoretical physics is based on the group-theoretical approach which generates the formalism of the principal fibre bundles and the instanton approach. The latter is based on the Pontrjagin's degree of map theorem and this theorem is the original ``bridge'' between homology and cohomology theories. The author plans to devote his two first lectures to fibre bundle theory: this is the foundation on which the modern physics rests -- the theor…
We reanalyze the perturbative radiative corrections to the Higgs mass within the Standard Model in the light of the Taylor-Lagrange renormalization scheme. This scheme naturally leads to completely finite corrections, depending on an arbitrary scale. The formulation avoids very large individual corrections to the Higgs mass. This illustrates the fact that the so-called fine-tuning problem in the Standard Model is just an artifact of the regularization scheme. It should therefore not lead to any physical interpretation in terms of the energy scale at which new physics should show up, nor in ter…
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