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Nonlinear Symmetries and Nonlinear Equations
The study of (nonlinear) dift"erential equations was S. Lie's motivation when he created what is now known as Lie groups and Lie algebras; nevertheless, although Lie group and algebra theory flourished and was applied to a number of dift"erent physical situations -up to the point that a lot, if not most, of current fun damental elementary particles physics is actually (physical interpretation of) group theory -the application of symmetry methods to dift"erential equations remained a sleeping beauty for many, many years. The main reason for this lies probably in a fact that is quite clear to any beginner in the field. Namely, the formidable comple:rity ofthe (algebraic, not numerical!) computations involved in Lie method. I think this does not account completely for this oblivion: in other fields of Physics very hard analytical computations have been worked through; anyway, one easily understands that systems of dOlens of coupled PDEs do not seem very attractive, nor a very practical computational tool.
Exploiting Symmetry in Applied and Numerical Analysis
Symmetry plays an important role in theoretical physics, applied analysis, classical differential equations, and bifurcation theory. Although numerical analysis has incorporated aspects of symmetry on an ad hoc basis, there is now a growing collection of numerical analysts who are currently attempting to use symmetry groups and representation theory as fundamental tools in their work. This book contains the proceedings of an AMS-SIAM Summer Seminar in Applied Mathematics, held in 1992 at Colorado State University. The seminar, which drew about 100 scientists from around the world, was intended to stimulate the systematic incorporation of symmetry and group theoretical concepts into numerical methods. The papers in this volume have been refereed and will not be published elsewhere.
Classical and Quantum Mechanics of Noncentral Potentials
Non-central forces have a wide variety of applications in classical and quantum mechanics as demonstrated in this book. The author emphasizes the study of time-dependent potentials, predominantly in two dimensions, without neglecting the quite well understood time-independent case. The construction of invariants in the classical case and the study of solutions to Schrödinger's equation, as well as a detailed presentation of various mathematical techniques are of main concern to the author. The book addresses theoretical physicists and mathematicians, but it will also be useful for electrical and mechanical engineers.
Modern Group Analysis: Advanced Analytical and Computational Methods in Mathematical Physics
On the occasion of the 150th anniversary of Sophus Lie, an International Work shop "Modern Group Analysis: advanced analytical and computational methods in mathematical physics" has been organized in Acireale (Catania, Sicily, October 27 31, 1992). The Workshop was aimed to enlighten the present state ofthis rapidly expanding branch of applied mathematics. Main topics of the Conference were: • classical Lie groups applied for constructing invariant solutions and conservation laws; • conditional (partial) symmetries; • Backlund transformations; • approximate symmetries; • group analysis of finite-difference equations; • problems of group classification; • software packages in gr...
Nonlinear Evolution Equations and Painlev Test
This book is an edited version of lectures given by the authors at a seminar at the Rand Afrikaans University. It gives a survey on the Painlev test, Painlev property and integrability. Both ordinary differential equations and partial differential equations are considered.
Applied Mathematics
The book is based on research presentations at the international conference, “Emerging Trends in Applied Mathematics: In the Memory of Sir Asutosh Mookerjee, S.N. Bose, M.N. Saha and N.R. Sen”, held at the Department of Applied Mathematics, University of Calcutta, during 12–14 February 2014. It focuses on various emerging and challenging topics in the field of applied mathematics and theoretical physics. The book will be a valuable resource for postgraduate students at higher levels and researchers in applied mathematics and theoretical physics. Researchers presented a wide variety of themes in applied mathematics and theoretical physics—such as emergent periodicity in a field of cha...
Deterministic Chaos in General Relativity
Nonlinear dynamical systems play an important role in a number of disciplines. The physical, biological, economic and even sociological worlds are comprised of com plex nonlinear systems that cannot be broken down into the behavior of their con stituents and then reassembled to form the whole. The lack of a superposition principle in such systems has challenged researchers to use a variety of analytic and numerical methods in attempts to understand the interesting nonlinear interactions that occur in the World around us. General relativity is a nonlinear dynamical theory par excellence. Only recently has the nonlinear evolution of the gravitational field described by the theory been tackled ...
Classical And Quantum Dissipative Systems (Second Edition)
Dissipative forces play an important role in problems of classical as well as quantum mechanics. Since these forces are not among the basic forces of nature, it is essential to consider whether they should be treated as phenomenological interactions used in the equations of motion, or they should be derived from other conservative forces. In this book we discuss both approaches in detail starting with the Stoke's law of motion in a viscous fluid and ending with a rather detailed review of the recent attempts to understand the nature of the drag forces originating from the motion of a plane or a sphere in vacuum caused by the variations in the zero-point energy. In the classical formulation, mathematical techniques for construction of Lagrangian and Hamiltonian for the variational formulation of non-conservative systems are discussed at length. Various physical systems of interest including the problem of radiating electron, theory of natural line width, spin-boson problem, scattering and trapping of heavy ions and optical potential models of nuclear reactions are considered and solved.
