Dynamics And Symmetry

This book contains the first systematic exposition of the global and local theory of dynamics equivariant with respect to a (compact) Lie group. Aside from general genericity and normal form theorems on equivariant bifurcation, it describes many general families of examples of equivariant bifurcation and includes a number of novel geometric techniques, in particular, equivariant transversality. This important book forms a theoretical basis of future work on equivariant reversible and Hamiltonian systems.This book also provides a general and comprehensive introduction to codimension one equivariant bifurcation theory. In particular, it includes the bifurcation theory developed with Roger Richardson on subgroups of reflection groups and the Maximal Isotropy Subgroup Conjecture. A number of general results are also given on the global theory. Introductory material on groups, representations and G-manifolds are covered in the first three chapters of the book. In addition, a self-contained introduction of equivariant transversality is given, including necessary results on stratifications as well as results on equivariant jet transversality developed by Edward Bierstone.

Whenever systems are governed by continuous chains of causes and effects, their behavior exhibits the consequences of dynamical symmetries, many of them far from obvious. Dynamical Symmetry introduces the reader to Sophus Lie's discoveries of the connections between differential equations and continuous groups that underlie this observation. It develops and applies the mathematical relations between dynamics and geometry that result. Systematic methods for uncovering dynamical symmetries are described, and put to use. Much material in the book is new and some has only recently appeared in research journals. Though Lie groups play a key role in elementary particle physics, their connection with differential equations is more often exploited in applied mathematics and engineering. Dynamical Symmetry bridges this gap in a novel manner designed to help readers establish new connections in their own areas of interest. Emphasis is placed on applications to physics and chemistry. Applications to many of the other sciences illustrate both general principles and the ubiquitousness of dynamical symmetries.

A graduate level text based partly on lectures in geometry, mechanics, and symmetry given at Imperial College London, this book links traditional classical mechanics texts and advanced modern mathematical treatments of the subject.

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Topological defects formed at symmetry-breaking phase transitions play an important role in many different fields of physics. They appear in many condensed-matter systems at low temperature; examples include vortices in superfluid helium-4, a rich variety of defects in helium-3, quantized mag netic flux tubes in type-II superconductors, and disclination lines and other defects in liquid crystals. In cosmology, unified gauge theories of particle interactions suggest a sequence of phase transitions in the very early uni verse some of which may lead to defect formation. In astrophysics, defects play an important role in the dynamics of neutron stars. In 1997 the European Science Foundation started the scientific network "Topological defects" headed by Tom Kibble. This network has provided us with a unique opportunity of establishing a collaboration between the representatives of these very different branches of modern physics. The NATO-ASI (Advanced Study Institute), held in Les Houches in February 1999 thanks to the support of the Scientific Division of NATO, the European Science Foundation and the CNRS, represents a key event of this ESF network. It brought together participants from widely different fields, with diverse expertise and vocabulary, fostering the exchange of ideas. The lectures given by particle physicists, cosmologists and condensed matter physicists are the result of the fruitful collaborations established since 1997 between groups in several European countries and in the U.S.A.

This book is an expanded version of a Master Class on the symmetric bifurcation theory of differential equations given by the author at the University of Twente in 1995. The notes cover a wide range of recent results in the subject, and focus on the dynamics that can appear in the generic bifurcation theory of symmetric differential equations. Many of the results and examples in the book are new and have not been previously published. The first four chapters contain an accessible presentation of the fundamental work by Field and Richardson on symmetry breaking and the Maximal Isotropy Subgroup Conjecture. The remainder of the book focuses on recent research of the author and includes chapters on the invariant sphere theorem, coupled cell systems, heteroclinic cycles , equivariant transversality, and an Appendix (with Xiaolin Peng) giving a new low dimensional counterexample to the converse of the Maximal Isotropy Subgroup Conjecture. The chapter on coupled cell systems includes a weath of new examples of 'cycling chaos'. The chapter on equivariant transversality is introductory and centres on an extended discussion of an explicit system of four coupled nonlinear oscillators. The style and format of the original lectures has largely been maintained and the notes include over seventy exercises *with hints for solutions and suggestions kfor further reading). In general terms, the notes are directed at mathematicians and aplied scientists working in the field of bifurcation theory who wish to learn about some of the latest developments and techniques in equivariant bifurcation theory. The notes are relatively self-contained and are structured so that they can form the basis for a graduate level course in equivariant bifurcation theory.

Serious visual artists can now easily understand and apply the secret geometry that masters used to create remarkable art. Superior mathematical skills aren’t required because there are hundreds of excellent step-by-step diagrams to explain everything with simplicity. Learn how the ancient and modern masters used dynamic symmetry to promote unity, movement, rhythm, and strength. These qualities, along with many others, allowed their art to have visual clarity, impact, and stand the test of time. This is an essential book for painters, photographers, sculptors, and cinematographers that hold composition and design with a high priority. For far too long, artists have been stuck with the basic tools of artistic composition, like the rule of thirds and leading lines. Unfortunately, we’re incapable of reaching the master level if all we know are the basics. Powerful tools like dynamic symmetry and other composition techniques have been kept a secret from all of us. It’s time to learn of them, push past any plateau that stands in our way, and finally unlock our true potential!

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

Is design intuitive or is it consciously and methodically worked out? Are there basic rules governing design that, when learned, will facilitate the creative process? These questions have been asked by artists, art historians, and art critics throughout the ages. Convinced that design was not purely instinctive, Jay Hambidge (1867–1924) spent much of his life searching for the technical bases of design. He found his answer in dynamic symmetry, one of the most provocative and stimulating theories in art history. Hambidge's study of Greek art convinced him that the secret of the beauty of Greek design was in the conscious use of dynamic symmetry — the law of natural design based upon the symmetry of growth in man and in plants. But Hambidge, who was not only a theoretician but also a practicing artist, did much more than analyze classical art and its principles of design: he worked out a series of root rectangles that the artist, using the simple mathematics supplied in this book, can easily follow and apply in his own work. Originally published as a series of lessons in Hambidge's magazine, The Diagonal, this engrossing book explains all the basic principles of dynamic symmetry. Part I sets forth the fundamental rectangles with their simple divisions based on the proportioning law found in nature; Part II explains compound rectangles, many of which were taken from or suggested by analysis of objects of Greek art. Whether read for its historical importance in art theory, for its illuminating insights into Greek art, or for its practical value to today's artists and commercial designers, The Elements of Dynamic Symmetry has much to offer anyone who is interested in the principle of design.

Describes methods revealing the structures and dynamics of turbulence for engineering, physical science and mathematics researchers working in fluid dynamics.