# Search Results for "geometric-mechanics-and-symmetry-the-peyresq-lectures-london-mathematical-society-lecture-note-series"

## Geometric Mechanics and Symmetry

*The Peyresq Lectures*

**Author**: James Montaldi,Tudor Ratiu**Publisher:**Cambridge University Press**ISBN:**9780521539579**Category:**Mathematics**Page:**402**View:**3588

Geometric mechanics lies on the border of pure and applied mathematics and incorporates such disciplines as differential geometry, Hamiltonian mechanics and integrable systems. The editors organised a summer school on Geometric Mechanics and Symmetry from which the main courses have been written up and published here. The book was written with a significant input from the participants at the conference. This means that the lecture notes are thoroughly geared towards the needs of a graduate student and take great care to explain concepts at the correct level.

## Geometric Mechanics: Dynamics and symmetry

**Author**: Darryl D. Holm**Publisher:**Imperial College Press**ISBN:**1848161956**Category:**Mathematics**Page:**354**View:**9025

Advanced undergraduate and graduate students in mathematics, physics and engineering.

## Geometric Mechanics

*Part I: Dynamics and Symmetry*

**Author**: Darryl D Holm**Publisher:**World Scientific Publishing Company**ISBN:**1911298658**Category:**Technology & Engineering**Page:**468**View:**2168

See also GEOMETRIC MECHANICS — Part II: Rotating, Translating and Rolling (2nd Edition) This textbook introduces the tools and language of modern geometric mechanics to advanced undergraduates and beginning graduate students in mathematics, physics and engineering. It treats the fundamental problems of dynamical systems from the viewpoint of Lie group symmetry in variational principles. The only prerequisites are linear algebra, calculus and some familiarity with Hamilton's principle and canonical Poisson brackets in classical mechanics at the beginning undergraduate level. The ideas and concepts of geometric mechanics are explained in the context of explicit examples. Through these examples, the student develops skills in performing computational manipulations, starting from Fermat's principle, working through the theory of differential forms on manifolds and transferring these ideas to the applications of reduction by symmetry to reveal Lie–Poisson Hamiltonian formulations and momentum maps in physical applications. The many Exercises and Worked Answers in the text enable the student to grasp the essential aspects of the subject. In addition, the modern language and application of differential forms is explained in the context of geometric mechanics, so that the importance of Lie derivatives and their flows is clear. All theorems are stated and proved explicitly. The organisation of the first edition has been preserved in the second edition. However, the substance of the text has been rewritten throughout to improve the flow and to enrich the development of the material. In particular, the role of Noether's theorem about the implications of Lie group symmetries for conservation laws of dynamical systems has been emphasised throughout, with many applications. Contents: Fermat's Ray Optics:Fermat's principleHamiltonian formulation of axial ray opticsHamiltonian form of optical transmissionAxisymmetric invariant coordinatesGeometry of invariant coordinatesSymplectic matricesLie algebrasEquilibrium solutionsMomentum mapsLie–Poisson bracketsDivergenceless vector fieldsGeometry of solution behaviourGeometric ray optics in anisotropic mediaTen geometrical features of ray opticsNewton, Lagrange, Hamilton and the Rigid Body:NewtonLagrangeHamiltonRigid-body motionSpherical pendulumLie, Poincaré, Cartan: Differential Forms:Poincaré and symplectic manifoldsPreliminaries for exterior calculusDifferential forms and Lie derivativesLie derivativeFormulations of ideal fluid dynamicsHodge star operator on ℝ3Poincaré's lemma: Closed vs exact differential formsEuler's equations in Maxwell formEuler's equations in Hodge-star form in ℝ4Resonances and S1 Reduction:Dynamics of two coupled oscillators on ℂ2The action of SU(2) on ℂ2Geometric and dynamic S1 phasesKummer shapes for n:m resonancesOptical travelling-wave pulsesElastic Spherical Pendulum:Introduction and problem formulationEquations of motionReduction and reconstruction of solutionsMaxwell-Bloch Laser-Matter Equations:Self-induced transparencyClassifying Lie–Poisson Hamiltonian structures for real-valued Maxwell–Bloch systemReductions to the two-dimensional level sets of the distinguished functionsRemarks on geometric phasesEnhanced Coursework:Problem formulations and selected solutionsIntroduction to oscillatory motionPlanar isotropic simple harmonic oscillator (PISHO)Complex phase space for two oscillatorsTwo-dimensional resonant oscillatorsA quadratically nonlinear oscillatorLie derivatives and differential formsExercises for Review and Further Study:The reduced Kepler problem: Newton (1686)Hamiltonian reduction by stagesℝ3 bracket for the spherical pendulumMaxwell–Bloch equationsModulation equationsThe Hopf map2:1 resonant oscillatorsA steady Euler fluid flowDynamics of vorticity gradientThe C Neumann problem (1859) Readership: Advanced undergraduate and graduate students in mathematics, physics and engineering; non-experts interested in geometric mechanics, dynamics and symmetry.

## Geometry, Mechanics, and Dynamics

*The Legacy of Jerry Marsden*

**Author**: Dong Eui Chang,Darryl D. Holm,George Patrick,Tudor Ratiu**Publisher:**Springer**ISBN:**1493924419**Category:**Mathematics**Page:**506**View:**450

This book illustrates the broad range of Jerry Marsden’s mathematical legacy in areas of geometry, mechanics, and dynamics, from very pure mathematics to very applied, but always with a geometric perspective. Each contribution develops its material from the viewpoint of geometric mechanics beginning at the very foundations, introducing readers to modern issues via illustrations in a wide range of topics. The twenty refereed papers contained in this volume are based on lectures and research performed during the month of July 2012 at the Fields Institute for Research in Mathematical Sciences, in a program in honor of Marsden's legacy. The unified treatment of the wide breadth of topics treated in this book will be of interest to both experts and novices in geometric mechanics. Experts will recognize applications of their own familiar concepts and methods in a wide variety of fields, some of which they may never have approached from a geometric viewpoint. Novices may choose topics that interest them among the various fields and learn about geometric approaches and perspectives toward those topics that will be new for them as well.

## International Journal of Applied Mathematics & Statistics

**Author**: N.A**Publisher:**N.A**ISBN:**N.A**Category:**Applied mathematics**Page:**N.A**View:**7774

## Geometric Mechanics: Rotating, translating and rolling

**Author**: Darryl D. Holm**Publisher:**N.A**ISBN:**9781848167759**Category:**Geometry, Differential**Page:**N.A**View:**5697

Advanced undergraduate and graduate students in mathematics, physics and engineering.

## Surveys in Combinatorics

*Invited Papers for the ... British Combinatorial Conference*

**Author**: Anthony Hilton,John Talbot**Publisher:**N.A**ISBN:**N.A**Category:**Combinatorial analysis**Page:**N.A**View:**6665

## Progress and Challenges in Dynamical Systems

*Proceedings of the International Conference Dynamical Systems: 100 Years after Poincaré, September 2012, Gijón, Spain*

**Author**: Santiago Ibáñez,Jesús S. Pérez del Río,Antonio Pumariño,J. Ángel Rodríguez**Publisher:**Springer Science & Business Media**ISBN:**3642388302**Category:**Mathematics**Page:**411**View:**7800

This book contains papers based on talks given at the International Conference Dynamical Systems: 100 years after Poincaré held at the University of Oviedo, Gijón in Spain, September 2012. It provides an overview of the state of the art in the study of dynamical systems. This book covers a broad range of topics, focusing on discrete and continuous dynamical systems, bifurcation theory, celestial mechanics, delay difference and differential equations, Hamiltonian systems and also the classic challenges in planar vector fields. It also details recent advances and new trends in the field, including applications to a wide range of disciplines such as biology, chemistry, physics and economics. The memory of Henri Poincaré, who laid the foundations of the subject, inspired this exploration of dynamical systems. In honor of this remarkable mathematician, theoretical physicist, engineer and philosopher, the authors have made a special effort to place the reader at the frontiers of current knowledge in the discipline.

## Geometric Mechanics and Symmetry

*From Finite to Infinite Dimensions*

**Author**: Darryl D. Holm,Tanya Schmah,Cristina Stoica**Publisher:**Oxford University Press**ISBN:**0199212902**Category:**Mathematics**Page:**515**View:**3350

Geometric Mechanics and Symmetry is a friendly and fast-paced introduction to the geometric approach to classical mechanics, suitable for a one- or two- semester course for beginning graduate students or advanced undergraduates. It fills a gap between traditional classical mechanics texts and advanced modern mathematical treatments of the subject.The modern geometric approach illuminates and unifies manyseemingly disparate mechanical problems from several areas of science and engineering. In particular, the book concentrates on the similarities between finite-dimensional rigid body motion and infinite-dimensional systems such asfluid flow. The illustrations and examples, together with a large number of exercises, both solved and unsolved, make the book particularly useful.

## Geometric Mechanics: Rotating, translating and rolling

**Author**: Darryl D. Holm**Publisher:**Imperial College Press**ISBN:**1848161557**Category:**Science**Page:**294**View:**3513

Introduces the tools and language of modern geometric mechanics to advanced undergraduate and beginning graduate students in mathematics, physics, and engineering. This book treats the dynamics of rotating, spinning and rolling rigid bodies from a geometric viewpoint, by formulating their solutions as coadjoint motions generated by Lie groups.

## Poisson Structures

**Author**: Camille Laurent-Gengoux,Anne Pichereau,Pol Vanhaecke**Publisher:**Springer Science & Business Media**ISBN:**3642310907**Category:**Mathematics**Page:**464**View:**2470

Poisson structures appear in a large variety of contexts, ranging from string theory, classical/quantum mechanics and differential geometry to abstract algebra, algebraic geometry and representation theory. In each one of these contexts, it turns out that the Poisson structure is not a theoretical artifact, but a key element which, unsolicited, comes along with the problem that is investigated, and its delicate properties are decisive for the solution to the problem in nearly all cases. Poisson Structures is the first book that offers a comprehensive introduction to the theory, as well as an overview of the different aspects of Poisson structures. The first part covers solid foundations, the central part consists of a detailed exposition of the different known types of Poisson structures and of the (usually mathematical) contexts in which they appear, and the final part is devoted to the two main applications of Poisson structures (integrable systems and deformation quantization). The clear structure of the book makes it adequate for readers who come across Poisson structures in their research or for graduate students or advanced researchers who are interested in an introduction to the many facets and applications of Poisson structures.

## The Symmetry Perspective

*From Equilibrium to Chaos in Phase Space and Physical Space*

**Author**: Martin Golubitsky,Ian Stewart**Publisher:**Birkhäuser**ISBN:**3034881673**Category:**Mathematics**Page:**325**View:**1128

The framework of ‘symmetry’ provides an important route between the abstract theory and experimental observations. The book applies symmetry methods to dynamical systems, focusing on bifurcation and chaos theory. Its exposition is organized around a wide variety of relevant applications. From the reviews: "[The] rich collection of examples makes the book...extremely useful for motivation and for spreading the ideas to a large Community."--MATHEMATICAL REVIEWS

## Global Analysis of Dynamical Systems

*Festschrift dedicated to Floris Takens for his 60th birthday*

**Author**: H.W Broer,B Krauskopf,Gert Vegter**Publisher:**CRC Press**ISBN:**9781420034288**Category:**Mathematics**Page:**464**View:**7439

Contributed by close colleagues, friends, and former students of Floris Takens, Global Analysis of Dynamical Systems is a liber amicorum dedicated to Takens for his 60th birthday. The first chapter is a reproduction of Takens's 1974 paper "Forced oscillators and bifurcations" that was previously available only as a preprint of the University of Utrecht. Among other important results, it contains the unfolding of what is now known as the Bogdanov-Takens bifurcation. The remaining chapters cover topics as diverse as bifurcation theory, Hamiltonian mechanics, homoclinic bifurcations, routes to chaos, ergodic theory, renormalization theory, and time series analysis. In its entirety, the book bears witness to the influence of Takens on the modern theory of dynamical systems and its applications. This book is a must-read for anyone interested in this active and exciting field.

## From Calculus to Cohomology

*De Rham Cohomology and Characteristic Classes*

**Author**: Ib H. Madsen,Jxrgen Tornehave**Publisher:**Cambridge University Press**ISBN:**9780521589567**Category:**Mathematics**Page:**286**View:**7843

An introductory textbook on cohomology and curvature with emphasis on applications.

## Applications of Contact Geometry and Topology in Physics

**Author**: Arkady Leonidovich Kholodenko**Publisher:**World Scientific**ISBN:**9814412090**Category:**Mathematics**Page:**492**View:**7853

Although contact geometry and topology is briefly discussed in V I Arnol''d''s book Mathematical Methods of Classical Mechanics (Springer-Verlag, 1989, 2nd edition), it still remains a domain of research in pure mathematics, e.g. see the recent monograph by H Geiges An Introduction to Contact Topology (Cambridge U Press, 2008). Some attempts to use contact geometry in physics were made in the monograph Contact Geometry and Nonlinear Differential Equations (Cambridge U Press, 2007). Unfortunately, even the excellent style of this monograph is not sufficient to attract the attention of the physics community to this type of problems. This book is the first serious attempt to change the existing status quo. In it we demonstrate that, in fact, all branches of theoretical physics can be rewritten in the language of contact geometry and topology: from mechanics, thermodynamics and electrodynamics to optics, gauge fields and gravity; from physics of liquid crystals to quantum mechanics and quantum computers, etc. The book is written in the style of famous Landau-Lifshitz (L-L) multivolume course in theoretical physics. This means that its readers are expected to have solid background in theoretical physics (at least at the level of the L-L course). No prior knowledge of specialized mathematics is required. All needed new mathematics is given in the context of discussed physical problems. As in the L-L course some problems/exercises are formulated along the way and, again as in the L-L course, these are always supplemented by either solutions or by hints (with exact references). Unlike the L-L course, though, some definitions, theorems, and remarks are also presented. This is done with the purpose of stimulating the interest of our readers in deeper study of subject matters discussed in the text.

## Mathematical Aspects of Natural Dynamos

**Author**: Emmanuel Dormy,Andrew M. Soward**Publisher:**CRC Press**ISBN:**1420055267**Category:**Science**Page:**504**View:**6479

Although the origin of Earth's and other celestial bodies' magnetic fields remains unknown, we do know that the motion of electrically conducting fluids generates and maintains these fields, forming the basis of magnetohydrodynamics (MHD) and, to a larger extent, dynamo theory. Answering the need for a comprehensive, interdisciplinary introduction to this area, Mathematical Aspects of Natural Dynamos provides a foundation in dynamo theory before moving on to modeling aspects of natural dynamos. Bringing together eminent international contributors, the book first introduces governing equations, outlines the kinematic dynamo theory, covers nonlinear effects, including amplitude saturation and polarity reversals, and discusses fluid dynamics. After establishing this base, the book describes the Earth's magnetic field and the current understanding of its characteristics. Subsequent chapters examine other planets in our solar system and the magnetic field of stars, including the sun. The book also addresses dynamo action on the large scale of galaxies, presents modeling experiments of natural dynamos, and speculates about future research directions. After reading this well-illustrated, thorough, and unified exploration, you will be well prepared to embark on your own journey through this fascinating area of research.

## Lectures on Duflo Isomorphisms in Lie Algebra and Complex Geometry

**Author**: Damien Calaque,Carlo A. Rossi**Publisher:**European Mathematical Society**ISBN:**9783037190968**Category:**Mathematics**Page:**106**View:**1398

## Current Issues in Cosmology

**Author**: Jean-Claude Pecker,Jayant Narlikar**Publisher:**Cambridge University Press**ISBN:**9780521858984**Category:**Science**Page:**N.A**View:**8632

What are the current ideas describing the large-scale structure of the Universe? How do they relate to the observed facts? This book, first published in 2006, looks at both the strengths and weaknesses of the current big-bang model in explaining certain puzzling data. It arises from an international conference that brought together many of the world's leading players in cosmology. In addition to presenting individual talks, the proceedings of the resulting discussions are also recorded. Giving a comprehensive coverage of the expanding field of cosmology, this text will be valuable for graduate students and researchers in cosmology and theoretical astrophysics.

## Geometric Mechanics

*Part II: Rotating, Translating and Rolling*

**Author**: Darryl D Holm**Publisher:**World Scientific Publishing Company**ISBN:**1911299336**Category:**Mathematics**Page:**312**View:**6376

This textbook introduces the tools and language of modern geometric mechanics to advanced undergraduate and beginning graduate students in mathematics, physics, and engineering. It treats the dynamics of rotating, spinning and rolling rigid bodies from a geometric viewpoint, by formulating their solutions as coadjoint motions generated by Lie groups. The only prerequisites are linear algebra, multivariable calculus and some familiarity with Euler-Lagrange variational principles and canonical Poisson brackets in classical mechanics at the beginning undergraduate level. Variational calculus on tangent spaces of Lie groups is explained in the context of familiar concrete examples. Through these examples, the student develops skills in performing computational manipulations, starting from vectors and matrices, working through the theory of quaternions to understand rotations, and then transferring these skills to the computation of more abstract adjoint and coadjoint motions, Lie-Poisson Hamiltonian formulations, momentum maps and finally dynamics with nonholonomic constraints. The 120 Exercises and 55 Worked Answers help the student to grasp the essential aspects of the subject, and to develop proficiency in using the powerful methods of geometric mechanics. In addition, all theorems are stated and proved explicitly. The book's many examples and worked exercises make it ideal for both classroom use and self-study. Contents: GalileoNewton, Lagrange, HamiltonQuaternionsQuaternionic ConjugacySpecial Orthogonal GroupThe Special Euclidean GroupGeometric Mechanics on SE(3)Heavy Top EquationsThe Euler–Poincaré TheoremLie–Poisson Hamiltonian FormMomentum MapsRound Rolling Rigid Bodies Readership: Advanced undergraduate and graduate students in mathematics, physics and engineering; researchers interested in learning the basic ideas in the fields; non-experts interested in geometric mechanics, dynamics and symmetry.