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This book is devoted to explaining a wide range of applications of con tinuous symmetry groups to physically important systems of differential equations. Emphasis is placed on significant applications of group-theoretic methods, organized so that the applied reader can readily learn the basic computational techniques required for genuine physical problems. The first chapter collects together (but does not prove) those aspects of Lie group theory which are of importance to differential equations. Applications covered in the body of the book include calculation of symmetry groups of differential…mehr
This book is devoted to explaining a wide range of applications of con tinuous symmetry groups to physically important systems of differential equations. Emphasis is placed on significant applications of group-theoretic methods, organized so that the applied reader can readily learn the basic computational techniques required for genuine physical problems. The first chapter collects together (but does not prove) those aspects of Lie group theory which are of importance to differential equations. Applications covered in the body of the book include calculation of symmetry groups of differential equations, integration of ordinary differential equations, including special techniques for Euler-Lagrange equations or Hamiltonian systems, differential invariants and construction of equations with pre scribed symmetry groups, group-invariant solutions of partial differential equations, dimensional analysis, and the connections between conservation laws and symmetry groups. Generalizations of the basic symmetry group concept, and applications to conservation laws, integrability conditions, completely integrable systems and soliton equations, and bi-Hamiltonian systems are covered in detail. The exposition is reasonably self-contained, and supplemented by numerous examples of direct physical importance, chosen from classical mechanics, fluid mechanics, elasticity and other applied areas.
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Autorenporträt
Jeff Calder received his Ph.D. degree in applied and interdisciplinary mathematics from the University of Michigan under the guidance of Prof. Selim Esedoglu and Prof. Alfred Hero in 2014. Between 2014 and 2016 he was a Morrey Assistant Professor at the University of California, Berkeley, under the mentorship of Lawrence C. Evans and James Sethian. He has been on the faculty of the School of Mathematics at the University of Minnesota since 2016, full professor since 2025, where he has supervised 5 PhD students, 4 postdoctoral scholars, and a number of undergraduate and high school students on research projects. Calder's research interests lie in applied probability, numerical analysis, and partial differential equations, with a specific interest in applications to machine learning and data analysis. Calder has published over 50 articles in journals and conferences spanning pure and applied mathematics and related areas, and holds several patents. His research has been recognized with an NSF Career Award and Alfred P. Sloan Research Fellowship in 2020, a University of Minnesota McKnight Presidential Fellowship and Guillermo E. Borja Award in 2021, and he currently holds the Albert and Dorothy Marden Professorship in Mathematics (2023-2028). Peter J. Olver received his Ph.D. from Harvard University in 1976 under the guidance of Prof. Garrett Birkhoff. After being a Dickson Instructor at the University of Chicago and a postdoc at the University of Oxford, he has been on the faculty of the School of Mathematics at the University of Minnesota since 1980, and a full professor since 1985. He served as the Head of the Department from 2008 to 2020. He has supervised 23 Ph.D. students, and mentored over 30 postdocs, visiting students and scholars from around the world, as well as supervising numerous undergraduate research projects. He is a Fellow of the American Mathematical Society, the Society for Industrial and Applied Mathematics (SIAM), the Institute of Physics, UK, and the Asia-Pacific Artificial Intelligence Association (AAIA). Over the years, he has contributed to a wide range of fields, including symmetry and Lie theory, partial differential equations, the calculus of variations, mathematical physics, fluid mechanics, elasticity, quantum mechanics, Hamiltonian mechanics, geometric numerical methods, differential geometry, classical invariant theory, algebra, computer vision and image processing, anthropology, and beyond. He is the author of over 160 papers in refereed journals, and has given more than 500 invited lectures on his research at conferences, universities, colleges, and institutes throughout the world. He was named a "Highly Cited Researcher" by Thomson-ISI in 2003, and an inaugural "Highly Ranked Scholar" by ScholarGPS in 2024.. He has written 6 books, including the definitive text on Applications of Lie Groups to Differential Equations, and two additional undergraduate texts: Partial Differential Equations and Applied Linear Algebra, the latter coauthored with his wife, Chehrzad Shakiban.
Inhaltsangabe
1 Introduction to Lie Groups.- 1.1. Manifolds.- 1.2. Lie Groups.- 1.3. Vector Fields.- 1.4. Lie Algebras.- 1.5. Differential Forms.- Notes.- Exercises.- 2 Symmetry Groups of Differential Equations.- 2.1. Symmetries of Algebraic Equations.- 2.2. Groups and Differential Equations.- 2.3. Prolongation.- 2.4. Calculation of Symmetry Groups.- 2.5. Integration of Ordinary Differential Equations.- 2.6. Nondegeneracy Conditions for Differential Equations.- Notes.- Exercises.- 3 Group-Invariant Solutions.- 3.1. Construction of Group-Invariant Solutions.- 3.2. Examples of Group-Invariant Solutions.- 3.3. Classification of Group-Invariant Solutions.- 3.4. Quotient Manifolds.- 3.5. Group-Invariant Prolongations and Reduction.- Notes.- Exercises.- 4 Symmetry Groups and Conservation Laws.- 4.1. The Calculus of Variations.- 4.2. Variational Symmetries.- 4.3. Conservation Laws.- 4.4. Noether's Theorem.- Notes.- Exercises.- 5 Generalized Symmetries.- 5.1. Generalized Symmetries of Differential Equations.- 5.2. Récursion Operators, Master Symmetries and Formal Symmetries.- 5.3. Generalized Symmetries and Conservation Laws.- 5.4. The Variational Complex.- Notes.- Exercises.- 6 Finite-Dimensional Hamiltonian Systems.- 6.1. Poisson Brackets.- 6.2. Symplectic Structures and Foliations.- 6.3. Symmetries, First Integrals and Reduction of Order.- Notes.- Exercises.- 7 Hamiltonian Methods for Evolution Equations.- 7.1. Poisson Brackets.- 7.2. Symmetries and Conservation Laws.- 7.3. Bi-Hamiltonian Systems.- Notes.- Exercises.- References.- Symbol Index.- Author Index.
