Generalized Polar Coordinates On Lie Groups And Numerical Intergrators

This volume resulted from presentations given at the international “Brainstorming Workshop on New Developments in Discrete Mechanics, Geometric Integration and Lie–Butcher Series”, that took place at the Instituto de Ciencias Matemáticas (ICMAT) in Madrid, Spain. It combines overview and research articles on recent and ongoing developments, as well as new research directions. Why geometric numerical integration? In their article of the same title Arieh Iserles and Reinout Quispel, two renowned experts in numerical analysis of differential equations, provide a compelling answer to this question. After this introductory chapter a collection of high-quality research articles aim at exploring recent and ongoing developments, as well as new research directions in the areas of geometric integration methods for differential equations, nonlinear systems interconnections, and discrete mechanics. One of the highlights is the unfolding of modern algebraic and combinatorial structures common to those topics, which give rise to fruitful interactions between theoretical as well as applied and computational perspectives. The volume is aimed at researchers and graduate students interested in theoretical and computational problems in geometric integration theory, nonlinear control theory, and discrete mechanics.

During the past two decades representations of noncompact Lie groups and Lie algebras have been studied extensively, and their application to other branches of mathematics and to physical sciences has increased enormously. Several theorems which were proved in the abstract now carry definite mathematical and physical sig nificance. Several physical observations which were not understood before are now explained in terms of models based on new group-theoretical structures such as dy namical groups and Lie supergroups. The workshop was designed to bring together those mathematicians and mathematical physicists who are actively working in this broad spectrum of research and to provide them with the opportunity to present their recent results and to discuss the challenges facing them in the many problems that remain. The objective of the workshop was indeed well achieved. This book contains 31 lectures presented by invited participants attending the NATO Advanced Research Workshop held in San Antonio, Texas, during the week of January 3-8, 1993. The introductory article by the editors provides a brief review of the concepts underlying these lectures (cited by author [*]) and mentions some of their applications. The articles in the book are grouped under the following general headings: Lie groups and Lie algebras, Lie superalgebras and Lie supergroups, and Quantum groups, and are arranged in the order in which they are cited in the introductory article. We are very thankful to Dr.