Добірка наукової літератури з теми "Lagrangian families"

AbstractOne of the most fundamental problems in the study of Lagrangian submanifolds from a Riemannian geometric point of view is the classification of Lagrangian immersions of real-space forms into complex-space forms. In this article, we solve this problem for the most basic case; namely, we classify Lagrangian surfaces of constant curvature in the complex Euclidean plane $\mathbb{C}^2$. Our main result states that there exist 19 families of Lagrangian surfaces of constant curvature in $\mathbb{C}^2$. Twelve of the 19 families are obtained via Legendre curves. Conversely, Lagrangian surfaces of constant curvature in $\mathbb{C}^2$ can be obtained locally from the 19 families.

The Yang-Mills theory structure is based on group theory. It rules the symmetry relationship where the number of potential fields transforming under a same group must be equal to the number of group generators. It defines the group valued expression from where the corresponding non-abelian symmetry properties are derived. Nevertheless based on different origins as Kaluza-Klein, fibre bundles, supersymmetry, s-model , BRST and anti-BRST algorithm, counting degrees of freedom leads to a Yang-Mills extension under the existence of different potential fields rotating under a same single group. They establish for SU(N) the relationship where and is a flavor index, . Physically, it says that different Yang-Mills families can share a common symmetry group. They develop a whole non-abelian gauge theory. The effort in this work is to explore such non-abelian extension. First, to build up it on the so-called constructor basis where gauge symmetry is more available for expressing the corresponding fields strengths, Lagrangian and classical equations. After that, given that the physical fields are those associated to the poles of two-point Green functions, one derives the physical Lagrangian L written in the physical basis . A new physical Lagrangian associated to symmetry is generated. The meaning of Yang-Mills families appears. A symmetry of difference is realized. Where every quanta is distinguished from each other. It yields a quanta diversity associated to corresponding Yang-Mills families. There are N-spin 1 and N-spin 0 quanta separated by different quantum numbers through a whole N-dynamics. An extension to QCD becomes possible.

The two-dimensional inverse problem for first-order systems is analysed and a method to construct an affine Lagrangian for such systems is developed. The determination of such Lagrangians is based on the theory of the Jacobi multiplier for the system of differential equations. We illustrate our analysis with several examples of families of forces that are relevant in mechanics, on one side, and of some relevant biological systems, on the other.

Cette thèse est consacrée à l'étude des cycles algébriques dans les variétés hyper-Kähleriennes projectives et les variétés de Calabi-Yau strictes. Elle contribue à la compréhension des conjectures de Beauville et de Voisin sur les anneaux de Chow des variétés hyper-kählériennes projectives et des variétés de Calabi-Yau strictes. Elle étudie également certains invariants birationnels des variétés hyper-kählériennes projectives.La première partie de la thèse, parue dans Mathematische Zeitschrift [C. Bai, On Abel-Jacobi maps of Lagrangian families, Math. Z. 304, 34 (2023)] et présentée dans le chapitre 2, étudie si les sous-variétés lagrangiennes dans une variété hyper-kählérienne partageant la même classe cohomologique ont également la même classe de Chow. Nous étudions la notion de familles lagrangiennes et ses applications aux applications d'Abel-Jacobi associées. Nous adoptons une approche infinitésimale pour donner un critère de trivialité de l'application d'Abel-Jacobi d'une famille lagrangienne, et utilisons ce critère pour donner une réponse négative à la question précédente, ajoutant aux subtilités d'une conjecture de Voisin. Nous explorons également comment la maximalité de la variation des structures de Hodge sur la cohomologie de degré 1 de la famille lagrangienne implique la trivialité de l'application d'Abel-Jacobi. La deuxième partie de la thèse, à paraître dans International Mathematics Research Notices [C. Bai, On some birational invariants of hyper-Kähler manifolds, ArXiv: 2210.12455, to appear in International Mathematics Research Notices, 2024] et présentée dans le chapitre 3, étudie le degré d'irrationalité, la gonalité fibrante et le genre fibrant des variétés hyper-kählériennes projectives. Nous commençons par donner une légère amélioration d'un résultat de Voisin sur la borne inférieure du degré d'irrationalité des variétés hyper-kählériennes générales de Mumford-Tate. Nous étudions ensuite la relation entre les trois invariants birationnels susmentionnés pour les surfaces K3 projectives de nombre de Picard 1, rajoutant la compréhension sur une conjecture de Bastianelli, De Poi, Ein, Lazarsfeld, Ullery sur le comportement asymptotique du degré d'irrationalité des surfaces K3 projectives très générales. La troisième partie de la thèse, présentée dans le chapitre 4, étudie les applications de Voisin de dimension supérieure sur les variétés de Calabi-Yau strictes. Voisin a construit des applications auto-rationnelles de variétés de Calabi-Yau obtenues comme des variétés de r-plans dans des hypersurfaces cubiques de dimension adéquate. Cette application a été largement étudiée dans le cas r=1, qui est le cas de Beauville-Donagi. Dans les cas de dimensions supérieures, nous étudions d'abord l'action de l'application de Voisin sur les formes holomorphes. Nous démontrons ensuite la conjecture de Bloch généralisée pour l'action des applications de Voisin sur les groupes de Chow dans le cas de r=2. Enfin, via l'étude de l'application de Voisin, nous apportons des éléments de preuve à une conjecture de Voisin sur l'existence d'un 0-cycle spécial sur les variétés de Calabi-Yau strictes
This thesis is devoted to the study of algebraic cycles in projective hyper-Kähler manifolds and strict Calabi-Yau manifolds. It contributes to the understanding of Beauville's and Voisin's conjectures on the Chow rings of projective hyper-Kähler manifolds and strict Calabi-Yau manifolds. It also studies some birational invariants of projective hyper-Kähler manifolds.The first part of the thesis, appeared in Mathematische Zeitschrift [C. Bai, On Abel-Jacobi maps of Lagrangian families, Math. Z. 304, 34 (2023)] and presented in Chapter 2, studies whether the Lagrangian subvarieties in a hyper-Kähler manifold sharing the same cohomological class have the same Chow class as well. We study the notion of Lagrangian families and its associated Abel-Jacobi maps. We take an infinitesimal approach to give a criterion for the triviality of the Abel-Jacobi map of a Lagrangian family, and use this criterion to give a negative answer to the above question, adding to the subtleties of a conjecture of Voisin. We also explore how the maximality of the variation of the Hodge structures on the degree 1 cohomology the Lagrangian family implies the triviality of the Abel-Jacobi map. The second part of the thesis, to appear in International Mathematics Research Notices [C. Bai, On some birational invariants of hyper-Kähler manifolds, ArXiv: 2210.12455, to appear in International Mathematics Research Notices, 2024] and presented in Chapter 3, studies the degree of irrationality, the fibering gonality and the fibering genus of projective hyper-Kähler manifolds, with emphasis on the K3 surfaces case, en mettant l'accent sur le cas des surfaces K3. We first give a slight improvement of a result of Voisin on the lower bound of the degree of irrationality of Mumford-Tate general hyper-Kähler manifolds. We then study the relation of the above three birational invariants for projective K3 surfaces of Picard number 1, adding the understandinf of a conjecture of Bastianelli, De Poi, Ein, Lazarsfeld, Ullery on the asymptotic behavior of the degree of irrationality of very general projective K3 surfaces. The third part of the thesis, presented in Chapter 4, studies the higher dimensional Voisin maps on strict Calabi-Yau manifolds. Voisin constructed self-rational maps of Calabi-Yau manifolds obtained as varieties of r-planes in cubic hypersurfaces of adequate dimension. This map has been thoroughly studied in the case r=1, which is the Beauville-Donagi case. For higher dimensional cases, we first study the action of the Voisin map on the holomorphic forms. We then prove the generalized Bloch conjecture for the action of the Voisin maps on Chow groups for the case of r=2. Finally, via the study of the Voisin map, we provide evidence for a conjecture of Voisin on the existence of a special 0-cycle on strict Calabi-Yau manifolds

Orientador : Loir Afonso Moreira
Dissertação (mestrado) - Universidade Estadual de Campinas, Faculdade de Engenharia Mecanica
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Resumo: Em 1970, Ahmad et aI. apresentaram uma formulação, para a análise de placas e cascas, baseada na degeneração de um elemento sólido tridimensional, através da redução de sua dimensão na direção da espessura. O campo de deslocamento era, então, interpolado a partir das funções de forma da família Serendipity e, para a obtenção das características do elemento, utilizava-se integração numérica consistente. Os resultados obtidos por esse elemento foram excelentes para situações de placas e cascas moderadamente grossas, atendendo, portando, à teoria de Mindlin- Reissner ~ contudo, com a redução da espessura, o elemento tomava-se excessivamente rígido e os resultados não tendiam, como era de se esperar, àqueles da teoria clássica de Kirchho./f para placas e cascas finas. O objetivo deste trabalho é o estudo comparativo, na análise de estabilidade de placas e cascas, entre o desempenho, utilizando-se integração numérica consistente, do elemento finito isoparamétrico, quadrilateral, quadrático da família Serendipity e do elemento finito isoparamétrico, quadrilateral, quadrático da família Lagrangiana, obtidos, ambos, a partir da formulação de Ahmad. Pretende-se estabelecer, dentre os dois elementos, qual é o mais adequado para iniciar o desenvolvimento de um elemento finito subparamétrico do tipo hierárquico, baseado no conceito da aproximação p, visando retirar a característica de rigidez excessiva do elemento de Ahmad, para situações de placas e cascas finas
Abstract: In 1970, Ahmad et aI. presented a formulation for the analysis of plates and shells, based on the degeneration of a three-dimensional solid element, by reducing its dimention in the thickness direction. The displacement field was then interpolated through the shape function of Serendipity fami1y and, in order to achieve the element characteristics, consistent integration was used. The obtained results were excellent for moderately thick plates and shells aplications, being in concordance with the Mind/in-Reissner's theory; however, with the reduction of the thickness, the element became excessively rigid and the results did not tend to those ofthe Kirchhoff's theory for thin plates and shells. The purpose of this work is the comparative study inthestabilityanalysis of plates and shells concerning the performance, considering consistent numerical integration, of the isoparametric, quadrilateral, quadratic finite element of Serendipity fami1y and of the isoparametric, quadrilateral, quadratic finite element of Lagrangian fami1y, achieved, both, ftom the Ahmad' s formulation, one intends to estabilish, between both elements, which is the most adequa te to start the development of a subparametric hierarchical element, based on p aproximation concept, objectifying to remove the characteristic of excessive rigidy of the Ahmad' s elements for the situation of thin plates and shells
Mestrado
Mestre em Educação Física

Orientador: Loir Afonso Moreira
Dissertação (mestrado) - Universidade Estadual de Campinas, Faculdade de Engenharia Mecanica
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Resumo: Em 1970, Ahmad apresentou uma formulação, para análise de placas e cascas, baseada na degeneração de um elemento tridimensional, através da redução de sua dimensão na direção da espessura. O campo de deslocamento era, então, interpolado a partir das funções de forma da família Serendipity e, para a obtenção das características do elemento, utilizava-se integração numérica consistente. Os resultados obtidos por esse elemento foram excelentes para situações de placas e cascas moderadamente grossas, atendendo, portanto, à teoria de Mindlin-Reissner; contudo, com a redução da espessura, o elemento tomava-se, excessivamente, rígido e os resultados não tendiam, como era de se esperar, àqueles da teoria clássica de Kirchhoff para placas e cascas finas. O objetivo deste trabalho é o estudo comparativo, na análise dinâmica de placas e cascas, entre o desempenho, utilizando-se integração numérica consistente, do elemento finito isoparamétrico, quadrilateral, quadrático, da família Serendipity e do elemento finito isoparamétrico, quadrilateral, quadrático, da família Lagrangeana, obtidos, ambos, a partir da formulação de Ahmad; pretende-se estabelecer, dentre os dois elementos, qual é o mais adequado para iniciar o desenvolvimento de um elemento finito subparamétrico do tipo hierárquico, baseado no conceito da aproximação p, visando retirar a característica de rigidez excessiva do elemento de Ahmad, para situações de placas e cascas finas
Abstract: In 1970, Ahmad presented a fonnulation for the analysis of plates and shells, based on the degeneration of three-dimensional solid element, by reducing its dimension in the thickness direction. The displacement field was then interpolated through the shape functions of Serendipity family and, in order to achieve the element characteristics, consistent numerical integration was used. The obtained results were excellent for moderately thick plates and shells applications, being in accordance with the Mindlin-Reissner's theory; however, with the reduction ofthe thickness, the element became excessively rigid and the results did not tend to those of the KirchhofJ's theory for thin plates and shells. The purpose of this work is the comparative study in the dynamic analysis of plates and shells conceming the perfonnance, considering consistent numerical integration, of the isoparametric, quadrilateral, quadratic finite element of Serendipity family and of the isoparametric, quadrilateral, quadratic finite element of Lagrangean family, achieved, both, from the Ahmad's fonnulation; one intends to establish, between both elements, which is the most adequate to start the development of a subparametric hierarchical element, based on the p approximation concept, objectifying to remove the characteristic of excessive rigidity of the Ahmad's element for the situations of thin plates and shells
Mestrado
Mecanica dos Sólidos e Projeto Mecanico
Mestre em Engenharia Mecânica

Orientador: Loir Afonso Moreira
Dissertação (mestrado) - Universidade de Campinas, Faculdade de Engenharia Mecanica
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Resumo: Em 1970, Ahmad apresentou uma formulação, para a análise de placas e cascas, baseada na degeneração de um elemento sólido tridimensional, através da redução de sua dimensão na direção da espessura. O campo de deslocamento era, então, interpolado a partir das funções de forma da familia Serendipity e, para a obtenção das características do elemento, utilizava-se integração numérica consistente. Os resultados obtidos por este elemento foram excelentes para situações de placas e cascas moderadamente grossas, atendendo, portanto, à teoria de Reissner. Contudo, com a redução da espessura, o elemento tomava-se excessivamente rígido e os resultados não tendiam, como era de se esperar, àqueles da teoria clássica de Kirchhoff para placas e cascas finas. O objetivo deste trabalho é o estudo comparativo, na análise estática de placas e cascas, entre o desempenho, utilizando-se integração numérica consistente, do elemento finito isoparamétrico, quadrilateral, quadrático, da família Serendipity e do elemento isoparamétrico, quadrilateral, quadrático, da família Lagrangeana, obtidos, ambos, a partir da formulação de Ahmad; pretende-se estabelecer, dentre os dois elementos, qual é o mais adequado para iniciar o desenvolvimento de um elemento finito subparamétrico do tipo hierárquico, baseado no conceito da aproximação p, visando retirar a característica de rigidez excessiva do elemento de Ahmad, para as situações de placas e cascas finas
Abstract: ln 1970, Ahmad presented a formulation for the analysis of plates and shells, based on the degeneration of a three-dimensional solid element, by reducing its dimension in the thickness direction. The displacement field was then interpolated through the shape functions of Serendipity family and, in order to achieve the element characteristics, consistent numerical integration was used. The results were excellent for moderately thick plates and shells applications, in accordance with the Reissner's theory; however, with the reduction of the thickness, the element became excessively rigid and the results did not tend to those of the Kirchhoff's theory for thin plates and shells. The purpose of this work is the comparative study in the static analysis of plates and shells concerning the performance, considering consistent numerical integration, of the isoparametric, quadrilateral, quadratic finite element of Serendipity family and of the isoparametric, quadrilateral, quadratic tinite element of Lagrangean family, obtained, both, trom the Ahmad's formulation. The main purpose is to establish, between both elements, which is the most adequate to start the development of a subparametric hierarchical element,based on the p-approximation concept, removing the characteristic of excessive rigidity of the Ahmad's element for the situations of thin plates and shells
Mestrado
Mecanica dos Sólidos e Projeto Mecanico
Mestre em Engenharia Mecânica

Dans cette thèse nous étudions la dynamique engendrée par une famille de flots Hamiltoniens. Un tel système dynamique à plusieurs générateurs est aussi appelé 'polysystème'. Motivés par des questions liées au phénomène de la diffusion d'Arnold, notre objectif est de construire des trajectoires du polysystème qui relient deux régions lointaines de l'espace des phases. La thèse est divisée en trois parties.Dans la première partie, nous considérons le polysystème engendré par les flots discrétisés d'une famille d'Hamiltoniens Tonelli. En utilisant une approche variationnelle issue de la théorie KAM faible, nous donnons des conditions suffisantes pour l'existence des trajectoires souhaitées.Dans la deuxième partie, nous traitons le cas d'un polysystème engendré par un couple de flots Hamiltoniens à temps continu, dont l'étude rentre dans le cadre de la théorie géométrique du contrôle. Dans ce contexte, nous montrons dans certains cas la transitivité d'un polysystème générique, à l'aide du théorème de transversalité de Thom.La dernière partie de la thèse est dédiée à obtenir une nouvelle version du théorème de transversalité de Thom s'exprimant en termes d'ensembles rectifiables de codimension positive. Dans cette partie il n'est pas question de polysystèmes, ni d'Hamiltoniens. Néanmoins, les résultats obtenus ici sont utilisés dans la deuxième partie de la thèse

Les erreurs dimensionnelles des pièces d'une famille de pièces provoquent un mauvais placement initial de la pièce sur le montage affectant la qualité du produit final. Même si la pièce est positionnée correctement, la pièce s'écarte de cette position initiale à cause des forces externe d'usinage et de bridage et de la rigidité du montage. Dans cette thèse, un modèle analytique complet, composé d'un modèle cinématique et un modèle mécanique, ayant une configuration de type 3-2-1 des appuis est proposé. Le modèle cinématique remet la pièce, initialement mal placée dans la référence machine, sur la bonne position par les avancements axiales des six appuis en tenant compte de tout les éléments du montage considérés rigides. Cette pièce repositionnée se déplace à nouveau à partir de la position corrigée sous les forces de bridage et d'usinage. Le modèle mécanique calcule ce déplacement de la pièce en considérant les appuis et des brides élastiques. Le plateau cuboïde rigide, utilisé pour repositionner la pièce précisément, est également considéré élastique au niveau des contacts avec les appuis. Le comportement non-linéaire de la déformation des contacts est linéarisé en convergeant la déformation des appuis jusqu'à ce que la précision requise est atteinte. En utilisant l'hypothèse des petits déplacements et en considérant le frottement nul au niveau des contacts, On calcule le déplacement de la pièce, la déformation de chaque appui suivant l'énergie minimale, la matrice de rigidité et le comportement mécanique du système du montage grâce à la formalisation Lagrangienne. Le déplacement de la pièce est à nouveau compensé par l'avancement axial des six appuis calculés par le modèle cinématique.

This chapter discusses the notion of action-minimizing orbits. In particular, it defines the other two families of invariant sets, the so-called Aubry and Mañé sets. It explains their main dynamical and symplectic properties, comparing them with the results obtained in the preceding chapter for the Mather sets. The relation between these new invariant sets and the Mather sets is described. As a by-product, the chapter introduces the Mañé's potential, Peierls' barrier, and Mañé's critical value. It discusses their properties thoroughly. In particular, it highlights how this critical value is related to the minimal average action and describes these new concepts in the case of the simple pendulum.

This chapter extends the now familiar Lagrangian formulation to a field theory and covers elementary material in this new setting. The motion of systems with a very large number of degrees of freedom makes it necessary to specify an almost infinite number of discrete coordinates. It is possible to simplify the situation by taking the continuum limit, which replaces the individual coordinates with a continuous function that describes a displacement field, which assigns a displacement vector to each position the system could occupy relative to an equilibrium configuration. The field thus takes a point in the spacetime manifold and assigns it a value corresponding to whatever the field represents. In this chapter, many interdisciplinary examples are solved and pedagogical models are discussed. The chapter also discusses Lagrange density, the Lagrange field equation, instantons, the Klein–Gordon equation, Fourier transforms and the Korteweg–de Vries equation.

In this paper, an extended Lagrangian formalism for general class of dynamical systems with dissipative, non-potential fields is formulated with the aim to obtain invariants of motion for such systems. A new concept of umbra-time has been introduced for this extension. D’Alembert basic idea of allowing displacement, when the real time is frozen is conveniently expressed in the terms of umbra-time. This leads to a peculiar form of equations, which is termed as umbra-Lagrange’s equations. A variational or least action doctrine leading to the proposed form of equation is introduced, which is based on recursive minimization of functionals. The concept of umbra-time extends the classical manifold over which the system evolves. The extension of Noether’s theorem in this extended space has been presented. The idea of umbra time is then used to propose the concept of umbra-Hamiltonian, which is used along with the extended Noether’s theorem to get into the dynamics of the dynamical systems with symmetries. In the mathematical models of dynamical system, the equations for the system can be formulated in a systematic way from the bondgraph representation as bondgraph representation of a system may be constructed in a total abstraction from the mathematical models of the dynamical system. In present paper, bond graphs are conveniently used to arrive at umbra-Lagrangian of the system. As a case study, we present a dynamic analysis of an electro-mechanical system through the proposed extended Lagrangian Formulation. The major objective of this paper is an analysis of symmetries of an electro-mechanical system comprising of an externally and internally damped, symmetric, elastic rotor driven by a three-phase induction motor, for which the umbra-Lagrangian remains unchanged under two families of transformations. The behaviour of limiting dynamics is obtained and validated through simulation studies.

There is currently quite a lot of interest in spatial solitions1-3 planar optical wave guides. The first calculations, involving bright and dark solitons propagating in magneto-optic materials, will be reported. They follow up recent work of the authors, based on Lagrangian formalism. It will be shown that magneto-optic problems divide, roughly, into polar, longitudinal, and transverse configurations, according to the orientation of the magnetic field to the soliton propagation direction. The system consists of a thin nonlinear film, sandwiched between a linear, nonmagnetic, semi-infinite, cladding, and a semi-infinite, magnetic, linear substrate. It is assumed that the power levels are only sufficient to drive the film into a nonlinear state. The discussion will show that this is reasonable and that the longitudinal and polar cases are of maximum, strategic interest. Technically, both cases are controlled off-diagonal elements in the dielectric tensor, which are proportional to a parameter Q, which, when unsealed, lies between 10-2 and 10-4. Although these values seem to be quite small, in absolute terms they compete vigorously with the nonlinearity. The starting point is a derivation of a new family of coupled envelope equations; all within the familiar, slowly varying amplitude, weakly nonlinear, weakly guiding formalism. We then demonstrate, by a scaling technique, that there exists an important, magneto-optic, cross-coupling term, which can sometimes be dominant. New parameters will be introduced that will be used mathematically, and numerically, to show that, with readily available materials, an external magnetic field is a powerful control parameter. The weakness of the nonlinearity is in accord with experiment and means that the modal fields are unchanged during the accumulation of nonlinear phase and magnetic Kerr phase shifts. Both are represented, mathematically, within a Lagrangian model. In the simplest case, when the magnetisation is a constant, very simple mathematical results for bright and dark solitons flow from this novel theory. They are vindicated by the numerical experiments. Beams that are attractive to each other in the absence of a magnetic field can be turned against each other. Dark beam channels can be significantly modified and multiple beam interactions are shown to imply a new range of devices. In particular, polarisation-dependent gating devices with small beam control will be used as illustrations. A really interesting possibility is the use of a magnetisation distribution. It is possible to laser anneal doped YIG, for example, and freeze the magnetisation into different directions at different points along the boundary, thus yielding various forms of Q = Q(x).

Improvements in efficiency and performance of gas turbines require a better understanding of the internal cooling air system which provides the turbine blades with cooling air. With the increase of cooling air passing through the internal air system, a greater amount of air borne particles is transported to the film cooling holes at the turbine blade surface. In spite of their small size, these holes are critical for blockage. Blockage of only a few holes could have harmful effects on the cooling film surrounding the blade. As a result, a reduced mean time between maintenance or even unexpected operation faults of the gas turbine during operation could occure. Experience showed a complex interaction of cooling air under different flow conditions and its particle load. To get more familiar with all these influences and the system itself, a test rig has been built. With this test rig, the behaviour of particles in the internal cooling air system could be studied at realistic flow conditions compared to a modern, heavy duty gas turbine. It is possible to simulate different particle sizes and dust concentrations in the coolant air. The test rig has been designed to give information about the quantity of separated particles at various critical areas of the internal air system [1]. The operation of the test rig as well as analysis of particles in such a complex flow system bear many problems, addressed in the previous paper [1]. New measurements and analysis methods give new and more accurate results, which will be shown in this paper. Furthermore the inspection of the test rig shows dust deposits at unexpected positions of the flow path. Theoretical studies to characterize the flow behaviour of the disperse phase in a continuous fluid using Lagrangian Tracking were also performed. A comparison between the numerical solution and the measurements will be shown in the paper.