Geometric Dynamics
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Geometrical Dynamics of Complex Systems
Author: Vladimir G. Ivancevic
language: en
Publisher: Taylor & Francis
Release Date: 2006-01-18
Geometrical Dynamics of Complex Systems is a graduate-level monographic textbook. Itrepresentsacomprehensiveintroductionintorigorousgeometrical dynamicsofcomplexsystemsofvariousnatures. By'complexsystems', inthis book are meant high-dimensional nonlinear systems, which can be (but not necessarily are) adaptive. This monograph proposes a uni?ed geometrical - proachtodynamicsofcomplexsystemsofvariouskinds: engineering, physical, biophysical, psychophysical, sociophysical, econophysical, etc. As their names suggest, all these multi-input multi-output (MIMO) systems have something in common: the underlying physics. However, instead of dealing with the pop- 1 ular 'soft complexity philosophy', we rather propose a rigorous geometrical and topological approach. We believe that our rigorous approach has much greater predictive power than the soft one. We argue that science and te- nology is all about prediction and control. Observation, understanding and explanation are important in education at undergraduate level, but after that it should be all prediction and control. The main objective of this book is to show that high-dimensional nonlinear systems and processes of 'real life' can be modelled and analyzed using rigorous mathematics, which enables their complete predictability and controllability, as if they were linear systems. It is well-known that linear systems, which are completely predictable and controllable by de?nition - live only in Euclidean spaces (of various - mensions). They are as simple as possible, mathematically elegant and fully elaborated from either scienti?c or engineering side. However, in nature, no- ing is linear. In reality, everything has a certain degree of nonlinearity, which means: unpredictability, with subsequent uncontrollability.
Geometry and Topology in Hamiltonian Dynamics and Statistical Mechanics
Author: Marco Pettini
language: en
Publisher: Springer Science & Business Media
Release Date: 2007-06-14
Itisaspecialpleasureformetowritethisforewordforaremarkablebookbya remarkableauthor.MarcoPettiniisadeepthinker,whohasspentmanyyears probing the foundations of Hamiltonian chaos and statistical mechanics, in particular phase transitions, from the point of view of geometry and topology. Itisinparticularthequalityofmindoftheauthorandhisdeepphysical,as well as mathematical insights which make this book so special and inspiring. It is a “must” for those who want to venture into a new approach to old problems or want to use new tools for new problems. Although topology has penetrated a number of ?elds of physics, a broad participationoftopologyintheclari?cationandprogressoffundamentalpr- lems in the above-mentioned ?elds has been lacking. The new perspectives topology gives to the above-mentioned problems are bound to help in their clari?cation and to spread to other ?elds of science. The sparsity of geometric thinking and of its use to solve fundamental problems, when compared with purely analytical methods in physics, could be relieved and made highly productive using the material discussed in this book. It is unavoidable that the physicist reader may have then to learn some new mathematics and be challenged to a new way of thinking, but with the author as a guide, he is assured of the best help in achieving this that is presently available.
Global Formulations of Lagrangian and Hamiltonian Dynamics on Manifolds
This book provides an accessible introduction to the variational formulation of Lagrangian and Hamiltonian mechanics, with a novel emphasis on global descriptions of the dynamics, which is a significant conceptual departure from more traditional approaches based on the use of local coordinates on the configuration manifold. In particular, we introduce a general methodology for obtaining globally valid equations of motion on configuration manifolds that are Lie groups, homogeneous spaces, and embedded manifolds, thereby avoiding the difficulties associated with coordinate singularities. The material is presented in an approachable fashion by considering concrete configuration manifolds of increasing complexity, which then motivates and naturally leads to the more general formulation that follows. Understanding of the material is enhanced by numerous in-depth examples throughout the book, culminating in non-trivial applications involving multi-body systems. This book is written for a general audience of mathematicians, engineers, and physicists with a basic knowledge of mechanics. Some basic background in differential geometry is helpful, but not essential, as the relevant concepts are introduced in the book, thereby making the material accessible to a broad audience, and suitable for either self-study or as the basis for a graduate course in applied mathematics, engineering, or physics.