Numerical Modelling of Thin Elastic Solids in Contact
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Abstract Book
14th International Geometry Symposium 25-28 May 2016 ABSTRACT BOOK Pamukkale University Denizli - TURKEY 1 14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016 14th International Geometry Symposium ABSTRACT BOOK 1 14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016 Proceedings of the 14th International Geometry Symposium Edited By: Dr. Şevket CİVELEK Dr. Cansel YORMAZ E-Published By: Pamukkale University Department of Mathematics Denizli, TURKEY All rights reserved. No part of this publication may be reproduced in any material form (including photocopying or storing in any medium by electronic means or whether or not transiently or incidentally to some other use of this publication) without the written permission of the copyright holder. Authors of papers in these proceedings are authorized to use their own material freely. Applications for the copyright holder’s written permission to reproduce any part of this publication should be addressed to: Assoc. Prof. Dr. Şevket CİVELEK Pamukkale University Department of Mathematics Denizli, TURKEY Email: [email protected] 2 14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016 Proceedings of the 14th International Geometry Symposium May 25-28, 2016 Denizli, Turkey. Jointly Organized by Pamukkale University Department of Mathematics Denizli, Turkey 3 14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016 PREFACE This volume comprises the abstracts of contributed papers presented at the 14th International Geometry Symposium, 14IGS 2016 held on May 25-28, 2016, in Denizli, Turkey. 14IGS 2016 is jointly organized by Department of Mathematics, Pamukkale University, Denizli, Turkey. The sysposium is aimed to provide a platform for Geometry and its applications. -
Covariant Momentum Map for Non-Abelian Topological BF Field
Covariant momentum map for non-Abelian topological BF field theory Alberto Molgado1,2 and Angel´ Rodr´ıguez–L´opez1 1 Facultad de Ciencias, Universidad Autonoma de San Luis Potosi Campus Pedregal, Av. Parque Chapultepec 1610, Col. Privadas del Pedregal, San Luis Potosi, SLP, 78217, Mexico 2 Dual CP Institute of High Energy Physics, Colima, Col, 28045, Mexico E-mail: [email protected], [email protected] Abstract. We analyze the inherent symmetries associated to the non-Abelian topological BF theory from the geometric and covariant perspectives of the Lagrangian and the multisymplectic formalisms. At the Lagrangian level, we classify the symmetries of the theory as natural and Noether symmetries and construct the associated Noether currents, while at the multisymplectic level the symmetries of the theory arise as covariant canonical transformations. These transformations allowed us to build within the multisymplectic approach, in a complete covariant way, the momentum maps which are analogous to the conserved Noether currents. The covariant momentum maps are fundamental to recover, after the space plus time decomposition of the background manifold, not only the extended Hamiltonian of the BF theory but also the generators of the gauge transformations which arise in the instantaneous Dirac-Hamiltonian analysis of the first-class constraint structure that characterizes the BF model under study. To the best of our knowledge, this is the first non-trivial physical model associated to General Relativity for which both natural and Noether symmetries have been analyzed at the multisymplectic level. Our study shed some light on the understanding of the manner in which the generators of gauge transformations may be recovered from the multisymplectic formalism for field theory. -
Lagrangian–Hamiltonian Unified Formalism for Field Theory
JOURNAL OF MATHEMATICAL PHYSICS VOLUME 45, NUMBER 1 JANUARY 2004 Lagrangian–Hamiltonian unified formalism for field theory Arturo Echeverrı´a-Enrı´quez Departamento de Matema´tica Aplicada IV, Edificio C-3, Campus Norte UPC, C/Jordi Girona 1, E-08034 Barcelona, Spain Carlos Lo´peza) Departamento de Matema´ticas, Campus Universitario. Fac. Ciencias, 28871 Alcala´ de Henares, Spain Jesu´s Marı´n-Solanob) Departamento de Matema´tica Econo´mica, Financiera y Actuarial, UB Avenida Diagonal 690, E-08034 Barcelona, Spain Miguel C. Mun˜oz-Lecandac) and Narciso Roma´n-Royd) Departamento de Matema´tica Aplicada IV, Edificio C-3, Campus Norte UPC, C/Jordi Girona 1, E-08034 Barcelona, Spain ͑Received 27 November 2002; accepted 15 September 2003͒ The Rusk–Skinner formalism was developed in order to give a geometrical unified formalism for describing mechanical systems. It incorporates all the characteristics of Lagrangian and Hamiltonian descriptions of these systems ͑including dynamical equations and solutions, constraints, Legendre map, evolution operators, equiva- lence, etc.͒. In this work we extend this unified framework to first-order classical field theories, and show how this description comprises the main features of the Lagrangian and Hamiltonian formalisms, both for the regular and singular cases. This formulation is a first step toward further applications in optimal control theory for partial differential equations. © 2004 American Institute of Physics. ͓DOI: 10.1063/1.1628384͔ I. INTRODUCTION In ordinary autonomous classical theories in mechanics there is a unified formulation of Lagrangian and Hamiltonian formalisms,1 which is based on the use of the Whitney sum of the ϭ ϵ ϫ ͑ tangent and cotangent bundles W TQ T*Q TQ QT*Q the velocity and momentum phase spaces of the system͒. -
Lectures on Classical Mechanics
Lectures on Classical Mechanics John C. Baez Derek K. Wise (version of September 7, 2019) i c 2017 John C. Baez & Derek K. Wise ii iii Preface These are notes for a mathematics graduate course on classical mechanics at U.C. River- side. I've taught this course three times recently. Twice I focused on the Hamiltonian approach. In 2005 I started with the Lagrangian approach, with a heavy emphasis on action principles, and derived the Hamiltonian approach from that. This approach seems more coherent. Derek Wise took beautiful handwritten notes on the 2005 course, which can be found on my website: http://math.ucr.edu/home/baez/classical/ Later, Blair Smith from Louisiana State University miraculously appeared and volun- teered to turn the notes into LATEX . While not yet the book I'd eventually like to write, the result may already be helpful for people interested in the mathematics of classical mechanics. The chapters in this LATEX version are in the same order as the weekly lectures, but I've merged weeks together, and sometimes split them over chapter, to obtain a more textbook feel to these notes. For reference, the weekly lectures are outlined here. Week 1: (Mar. 28, 30, Apr. 1)|The Lagrangian approach to classical mechanics: deriving F = ma from the requirement that the particle's path be a critical point of the action. The prehistory of the Lagrangian approach: D'Alembert's \principle of least energy" in statics, Fermat's \principle of least time" in optics, and how D'Alembert generalized his principle from statics to dynamics using the concept of \inertia force". -
Geometric Justification of the Fundamental Interaction Fields For
S S symmetry Article Geometric Justification of the Fundamental Interaction Fields for the Classical Long-Range Forces Vesselin G. Gueorguiev 1,2,* and Andre Maeder 3 1 Institute for Advanced Physical Studies, Sofia 1784, Bulgaria 2 Ronin Institute for Independent Scholarship, 127 Haddon Pl., Montclair, NJ 07043, USA 3 Geneva Observatory, University of Geneva, Chemin des Maillettes 51, CH-1290 Sauverny, Switzerland; [email protected] * Correspondence: [email protected] Abstract: Based on the principle of reparametrization invariance, the general structure of physically relevant classical matter systems is illuminated within the Lagrangian framework. In a straight- forward way, the matter Lagrangian contains background interaction fields, such as a 1-form field analogous to the electromagnetic vector potential and symmetric tensor for gravity. The geometric justification of the interaction field Lagrangians for the electromagnetic and gravitational interactions are emphasized. The generalization to E-dimensional extended objects (p-branes) embedded in a bulk space M is also discussed within the light of some familiar examples. The concept of fictitious accelerations due to un-proper time parametrization is introduced, and its implications are discussed. The framework naturally suggests new classical interaction fields beyond electromagnetism and gravity. The simplest model with such fields is analyzed and its relevance to dark matter and dark energy phenomena on large/cosmological scales is inferred. Unusual pathological behavior in the Newtonian limit is suggested to be a precursor of quantum effects and of inflation-like processes at microscopic scales. Citation: Gueorguiev, V.G.; Maeder, A. Geometric Justification of Keywords: diffeomorphism invariant systems; reparametrization-invariant matter systems; matter the Fundamental Interaction Fields lagrangian; homogeneous singular lagrangians; relativistic particle; string theory; extended objects; for the Classical Long-Range Forces. -
Rigid Body Dynamic Simulation with Line and Surface Contact Jiayin Xie, Student Member, IEEE, Nilanjan Chakraborty , Member, IEEE
1 Rigid Body Dynamic Simulation with Line and Surface Contact Jiayin Xie, Student Member, IEEE, Nilanjan Chakraborty , Member, IEEE, Abstract—In this paper, we develop a principled method to model line and surface contact with point contact (we call this point, equivalent contact point) that is consistent with physics- based models of surface (line) contact. Assuming that the set of contact points form a convex set, we solve the contact detection and dynamic simulation step simultaneously by formulating the problem as a mixed nonlinear complementarity problem. This allows us to simultaneously compute the equivalent contact point as well as the wrenches (forces and moments) at the equivalent contact point (consistent with the friction model) along with the Fig. 1: State-of-the-art dynamic simulation algorithms assume configuration and velocities of the rigid objects. Furthermore, that the contact between two objects is always a point contact we prove that the contact constraints of no inter-penetration (left). However, the contact may actually be a (middle) line between the objects is also satisfied. We present a geometri- cally implicit time-stepping scheme for dynamic simulation for or (right) surface contact. We develop a principled method to contacts between two bodies with convex contact area, which incorporate line and surface contact in rigid body dynamic includes line contact and surface contact. We prove that for simulation. surface and line contact, for any value of the velocity of center of mass of the object, there is a unique solution for contact point and contact wrench that satisfies the discrete-time equations of motion. -
Mathematical Structures in Physics
Mathematical Structures in Physics Kolleg Studienstiftung Christoph Schweigert Hamburg University Department of Mathematics Section Algebra and Number Theory and Center for Mathematical Physics (as of 12.03.2016) Contents 0 Introduction 1 1 Newtonian mechanics 2 1.1 Galilei space, equations of motion . 2 1.2 Dynamics of Newtonian systems . 8 1.3 Examples . 10 2 Lagrangian mechanics 14 2.1 Variational calculus . 14 2.2 Systems with constraints . 24 2.3 Lagrangian systems: jet bundles as the kinematical setup . 28 2.4 Lagrangian dynamics . 43 2.5 Symmetries and Noether identities . 48 2.6 Natural geometry . 57 3 Classical field theories 61 3.1 Maxwell's equations . 61 3.2 Special relativity . 69 3.4 Electrodynamics as a gauge theory . 77 3.5 General relativity . 85 4 Hamiltonian mechanics 89 4.1 (Pre-) symplectic manifolds . 89 4.2 Poisson manifolds and Hamiltonian systems . 99 4.3 Time dependent Hamiltonian dynamics . 105 4.4 The Legendre transform . 108 i 5 Quantum mechanics 115 5.1 Deformations . 115 5.2 Kinematical framework for quantum mechanics: C∗-algebras and states . 119 5.3 Composite systems and Bell's inequality . 130 5.4 Dynamics of quantum mechanical systems . 132 5.5 Quantization . 141 5.6 Symmetries in quantum mechanics . 150 5.7 Examples of quantum mechanical systems . 159 5.8 Quantum statistical mechanics and KMS states . 168 5.9 Perturbation theory . 170 5.10 Path integral methods . 170 6 A glimpse to quantum field theory 171 A Differentiable Manifolds 174 A.1 Definition of differentiable manifolds . 174 A.2 Tangent vectors and differentiation . -
Formulation and Preparation for Numerical Evaluation of Linear Complementarity Systems in Dynamics
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by RERO DOC Digital Library Multibody System Dynamics (2005) 13: 447–463 C Springer 2005 Formulation and Preparation for Numerical Evaluation of Linear Complementarity Systems in Dynamics CHRISTOPH GLOCKER and CHRISTIAN STUDER IMES - Center of Mechanics, ETH Zurich, CH-8092 Zurich, Switzerland; E-mail: [email protected] (Received 4 February 2004; accepted in revised form 19 May, 2004) Abstract. In this paper, we provide a full instruction on how to formulate and evaluate planar fric- tional contact problems in the spirit of non-smooth dynamics. By stating the equations of motion as an equality of measures, frictional contact reactions are taken into account by Lagrangian multipliers. Contact kinematics is formulated in terms of gap functions, and normal and tangential relative ve- locities. Associated frictional contact laws are stated as inclusions, incorporating impact behavior in form of Newtonian kinematic impacts. Based on this inequality formulation, a linear complementarity problem in standard form is presented, combined with Moreau’s time stepping method for numerical integration. This approach has been applied to the woodpecker toy, of which a complete parameter list and numerical results are given in the paper. Keywords: unilateral constraints, friction, impact, time stepping, linear complementary problem 1. Introduction During the last years, increasing interest in the behavior of dynamic systems with discontinuities has been observed in nonlinear dynamics, comprising in particular friction and impact problems. As research in the past was primarily focussed on systems with a single non-smooth interaction element, current trends point towards more and more complex situations, involving combined frictional and unilateral impact behavior and multi-contact problems. -
Noether's Theorem and Symmetry
Noether’s Theorem and Symmetry Edited by P.G.L. Leach and Andronikos Paliathanasis Printed Edition of the Special Issue Published in Symmetry www.mdpi.com/journal/symmetry Noether’s Theorem and Symmetry Noether’s Theorem and Symmetry Special Issue Editors P.G.L. Leach Andronikos Paliathanasis MDPI • Basel • Beijing • Wuhan • Barcelona • Belgrade Special Issue Editors P.G.L. Leach Andronikos Paliathanasis University of Kwazulu Natal & Durban University of Technology Durban University of Technology Republic of South Arica Cyprus Editorial Office MDPI St. Alban-Anlage 66 4052 Basel, Switzerland This is a reprint of articles from the Special Issue published online in the open access journal Symmetry (ISSN 2073-8994) from 2018 to 2019 (available at: https://www.mdpi.com/journal/symmetry/ special issues/Noethers Theorem Symmetry). For citation purposes, cite each article independently as indicated on the article page online and as indicated below: LastName, A.A.; LastName, B.B.; LastName, C.C. Article Title. Journal Name Year, Article Number, Page Range. ISBN 978-3-03928-234-0 (Pbk) ISBN 978-3-03928-235-7 (PDF) Cover image courtesy of Andronikos Paliathanasis. c 2020 by the authors. Articles in this book are Open Access and distributed under the Creative Commons Attribution (CC BY) license, which allows users to download, copy and build upon published articles, as long as the author and publisher are properly credited, which ensures maximum dissemination and a wider impact of our publications. The book as a whole is distributed by MDPI under the terms and conditions of the Creative Commons license CC BY-NC-ND. -
Symmetries and Gauge Symmetries in Multisymplectic First and Second
SYMMETRIES AND GAUGE SYMMETRIES IN MULTISYMPLECTIC FIRST AND SECOND-ORDER LAGRANGIAN FIELD THEORIES: ELECTROMAGNETIC AND GRAVITATIONAL FIELDS 1JORDI GASET *, 2NARCISO ROMAN´ -ROY † 1Department of Physics, Universitat Autonoma` de Barcelona, Bellaterra, Spain. 2Department of Mathematics, Universitat Politecnica` de Catalunya, Barcelona, Spain. July 20, 2021 Abstract Symmetries and, in particular, Cartan (Noether) symmetries and conserved quantities (conser- vation laws) are studied for the multisymplectic formulation of first and second order Lagrangian classical field theories. Noether-type theorems are stated in this geometric framework. The concept of gauge symmetry and its geometrical meaning are also discussed in this formulation. The results are applied to study Noether and gauge symmetries for the multisymplectic description of the elec- tromagnetic and the gravitational theory; in particular, the Einstein–Hilbert and the Einstein–Palatini approaches. Key words: 1st and 2nd-order Lagrangian field theories, Higher-order jet bundles, Multisymplectic forms, symmetries, gauge symmetries, conservation laws, Noether theorem, Hilbert-Einstein action, Einstein-Palatini approach. MSC 2020 codes: Primary: 53D42, 70S05, 83C05; Secondary: 35Q75, 35Q76, 53Z05, 70H50, 83C99. arXiv:2107.08846v1 [math-ph] 19 Jul 2021 Contents 1 Introduction 2 2 Lagrangian field theories in jet bundles 3 2.1 Higher-order jet bundles. Multivector fields in jet bundles...................... 3 2.2 First andsecond-orderLagrangianfieldtheories . ..................... 4 3 Symmetries, conservation laws and gauge symmetries 6 3.1 Symmetries and conserved quantities for Lagrangian fieldtheories ................. 6 3.2 Gauge symmetries, gauge vector fields and gauge equivalence ................... 9 4 Electromagnetic field 12 5 Gravitational field (General Relativity) 13 5.1 TheHilbert-Einsteinaction . ............... 13 5.2 The Einstein-Palatini action (metric-affine model) . -
Exomars Rover Mechanical Modeling with Siconos Jan Michalczyk, Maurice Br´Emond,Vincent Acary, Roger Pissard-Gibollet
Exomars Rover Mechanical Modeling with Siconos Jan Michalczyk, Maurice Br´emond,Vincent Acary, Roger Pissard-Gibollet To cite this version: Jan Michalczyk, Maurice Br´emond, Vincent Acary, Roger Pissard-Gibollet. Exomars Rover Mechanical Modeling with Siconos. [Technical Report] RT-0448, INRIA. 2014, pp.41. <hal- 01025785> HAL Id: hal-01025785 https://hal.inria.fr/hal-01025785 Submitted on 18 Jul 2014 HAL is a multi-disciplinary open access L'archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destin´eeau d´ep^otet `ala diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publi´esou non, lished or not. The documents may come from ´emanant des ´etablissements d'enseignement et de teaching and research institutions in France or recherche fran¸caisou ´etrangers,des laboratoires abroad, or from public or private research centers. publics ou priv´es. Exomars Rover Mechanical Modeling with Siconos Jan Michalczyk, Maurice Brémond, Vincent Acary, Roger Pissard-Gibollet TECHNICAL REPORT N° 448 July 2014 Project-Team Bipop ISSN 0249-0803 ISRN INRIA/RT--448--FR+ENG Exomars Rover Mechanical Modeling with Siconos Jan Michalczyk, Maurice Brémond, Vincent Acary, Roger Pissard-Gibollet Project-Team Bipop Technical Report n° 448 — July 2014 — 38 pages Abstract: This document contains specification and documentation of the tests performed on the exomars planetary rover model using Siconos software. Siconos is an open source scientific software targeted at modeling and simulating nonsmooth dynamical systems. The advantage of using Siconos for mechanical simulations is that it offers efficient friction-contact nonsmooth models. Performing mechanical simulations of the exomars rover appears to be of great importance with respect to verifying its design and estimating energy consumption. -
Oblique Frictional Impact of a Bar: Analysis and Comparison of Different Impact Laws
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by RERO DOC Digital Library Nonlinear Dynamics (2005) 41: 361–383 c Springer 2005 Oblique Frictional Impact of a Bar: Analysis and Comparison of Different Impact Laws MATTHIAS PAYR and CHRISTOPH GLOCKER∗ IMES – Center of Mechanics, ETH Zurich, 8092 Zurich, Switzerland; ∗Author for correspondence (e-mail: [email protected]; fax: +41-1-632-1145) (Received: 3 March 2004; accepted: 14 December 2004) Abstract. In this paper a basic, easily to multi-contact problems extendable, non-smooth approach is applied to analyze a bar striking an inelastic half-space. Coulomb contact is assumed and modeled by using set-valued Newtonian impact laws in normal as well as in tangential direction. The resulting linear complementarity problem contains all possible impact states and provides an instantaneous collision operator that respects all inequality constraints. This operator depends on the orientation of the bar and determines uniquely the post-impact velocities as functions of the pre-impact state. Different types of solutions may occur, including “stick” and “slip”. In this context, stick and slip have to be understood as the two cases characterized by the tangential impulsive force as an element of either the set-valued or of the single-valued domain of the friction law. Depending on the choice of parameters, sign reversal of the tangential contact velocity is possible. For certain inertia properties and initial conditions, the collision operator yields an impact, even for initially vanishing normal contact velocity. This phenomenon is well known as the Painlev´e paradox.