الصفحة 1
الصفحة 1
Metamorphoses of Hamiltonian Systems with Symmetries
Modern notions and important tools of classical mechanics are used in the study of concrete examples that model physically significant molecular and atomic systems. The parametric nature of these examples leads naturally to the study of the major qualitative changes of such systems (metamorphoses) as the parameters are varied. The symmetries of these systems, discrete or continuous, exact or approximate, are used to simplify the problem through a number of mathematical tools and techniques like normalization and reduction. The book moves gradually from finding relative equilibria using symmetry, to the Hamiltonian Hopf bifurcation and its relation to monodromy and, finally, to generalizations of monodromy.
Integrable Systems in Celestial Mechanics
This work presents a unified treatment of three important integrable problems relevant to both Celestial and Quantum Mechanics. Under discussion are the Kepler (two-body) problem and the Euler (two-fixed center) problem, the latter being the more complex and more instructive, as it exhibits a richer and more varied solution structure. Further, because of the interesting investigations by the 20th century mathematical physicist J.P. Vinti, the Euler problem is now recognized as being intimately linked to the Vinti (Earth-satellite) problem. Here the analysis of these problems is shown to follow a definite shared pattern yielding exact forms for the solutions. A central feature is the detailed treatment of the planar Euler problem where the solutions are expressed in terms of Jacobian elliptic functions, yielding analytic representations for the orbits over the entire parameter range.
Infinite dimensional algebras and quantum integrable systems
This volume presents the invited lectures of the workshop "Infinite Dimensional Algebras and Quantum Integrable Systems'' .ecent developments in the theory of infinite dimensional algebras and their applications to quantum integrable systems are reviewed by some of the leading experts in the field. The volume will be of interest to a broad audience from graduate students to researchers in mathematical physics and related fields.
Ernst Equation and Riemann Surfaces : Analytical and Numerical Methods
Exact solutions to Einstein`s equations have been useful for the understanding of general relativity in many respects. They have led to physical concepts as black holes and event horizons and helped to visualize interesting features of the theory. In addition they have been used to test the quality of various approximation methods and numerical codes. The most powerful solution generation methods are due to the theory of Integrable Systems. In the case of axisymmetric stationary spacetimes the Einstein equations are equivalent to the completely integrable Ernst equation. In this volume the solutions to the Ernst equation associated to Riemann surfaces are studied in detail and physical and mathematical aspects of this class are discussed both analytically and numerically.
Dissipative Solitons : From Optics to Biology and Medicine
The dissipative soliton concept is a fundamental extension of the concept of solitons in conservative and integrable systems. It includes ideas from three major sources, namely standard soliton theory developed since the 1960s, nonlinear dynamics theory, and Prigogine's ideas of systems far from equilibrium. These three sources also correspond to the three component parts of this novel paradigm. This book explains the above principles in detail and gives the reader various examples from optics, biology and medicine. These include laser systems, optical transmission lines, cortical networks, models of muscle contraction, localized vegetation structures and waves in brain tissues.
Dissipative Solitons
This volume is devoted to the exciting topic of dissipative solitons, i.e. pulses or spatially localised waves in systems exhibiting gain and loss. Examples are laser systems, nonlinear resonators and optical transmission lines. The physical principles and mathematical concepts are explained in a clear and concise way, suitable for students and young researchers. The similarities and differences in the notion of a soliton between dissipative systems and Hamiltonian and integrable systems are discussed, and many examples are given. The contributions are written by the world's leading experts in the field, making it a unique exposition of this emerging topic.
Darboux Transformations in Integrable Systems : Theory and their Applications to Geometry
This book presents the Darboux transformations in matrix form and provides purely algebraic algorithms for constructing the explicit solutions. A basis for using symbolic computations to obtain the explicit exact solutions for many integrable systems is established. Moreover, the behavior of simple and multi-solutions, even in multi-dimensional cases, can be elucidated clearly. The method covers a series of important equations such as various kinds of AKNS systems in R1+n, harmonic maps from 2-dimensional manifolds, self-dual Yang-Mills fields and the generalizations to higher dimensional case, theory of line congruences in three dimensions or higher dimensional space etc. All these cases are explained in detail.
Calculus and mechanics on two-point homogenous riemannian spaces
The present monograph gives a short and concise introduction to classical and quantum mechanics on two-point homogenous Riemannian spaces, with empahsis on spaces with constant curvature. Chapter 1-4 provide the basic notations from differential geometry for studying two-body dynamics in these spaces. Chapter 5 deals with the problem of finding explicitly invariant expressions for the two-body quantum Hamiltonian. Chapter 6 addresses one-body problems in a central potential. Chapter 7 studies the classical counterpart of the quantum system of chapter 5. Chapter 8 investigates some applications in the quantum realm, namely for the coulomb and oscillator potentials.
Bilinear integrable systems : From classical to quantum, continuous to discrete ; Proceedings of the NATO Advanced Research Workshop on Bilinear Integrable Systems: From Classical to Quantum, Continuous to Discrete St. Petersburg, Russia, 15-19 September 2002
Trained as a physicistin his home university Kyushu University, Professor Hirota earned his PhD in’61 at Northwestern University with Professor Siegert in the field of “QuantumStatistical mechanics”. He wrote a widely appreciated Doctoral dissertation on“Functional Integral representation of the grand partition function”. As a youngresearcher, he entered the RCA Company in Tokyo to do research on semi-conductor plasmas. Professor Hirota was led to model the Toda lattice as a non-linear networkof ladder-type LC circuits. The self-dual case led to equations very reminiscentof the Sine-Gordon equation, with much the same features (existence of onesoliton, soliton-soliton interaction, etc)
Algebraic Analysis of Differential Equations : from Microlocal Analysis to Exponential Asymptotics Festschrift in Honor of Takahiro Kawai Editors
This volume contains 23 articles on algebraic analysis of differential equations and related topics, most of which were presented as papers at the international conference "Algebraic Analysis of Differential Equations – from Microlocal Analysis to Exponential Asymptotics" at Kyoto University in 2005. Microlocal analysis and exponential asymptotics are intimately connected and provide powerful tools that have been applied to linear and non-linear differential equations as well as many related fields such as real and complex analysis, integral transforms, spectral theory, inverse problems, integrable systems, and mathematical physics. The articles contained here present many new results and ideas, providing interested researchers and students with valuable suggestions and instructive guidance for their work.
Advances in Discrete Differential Geometry
On a newly emerging field of discrete differential geometry and an excellent way to access this exciting area. It surveys the fascinating connections between discrete models in differential geometry and complex analysis, integrable systems and applications in computer graphics. The authors take a closer look at discrete models in differential geometry and dynamical systems. Their curves are polygonal, surfaces are made from triangles and quadrilaterals, and time is discrete. Nevertheless, the difference between the corresponding smooth curves, surfaces and classical dynamical systems with continuous time can hardly be seen. This is the paradigm of structure-preserving discretizations. Current advances in this field are stimulated to a large extent by its relevance for computer graphics and mathematical physics.
Advanced mathematical science for mobility society
The automotive industry has made steady progress in technological innovations under the names of Connected Autonomous-Shared-Electric (CASE) and Mobility as a Service (MaaS). Needless to say, mathematics and informatics are important to support such innovations. As the concept of cars and movement itself is diversifying, they are indispensable for grasping the essence of the future mobility society and building the foundation for the next generation. This book contains three main contents. 1. Mathematical models of flow 2. Mathematical methodsfor huge data and network analysis 3. Algorithm for mobility society The first one discusses mathematical models of pedestrian and traffic flow, as they are important for preventing accidents and achieving efficient transportation.
