Discrete Mechanics

The discrete vision of mechanics is based on the founding ideas of Galileo and the principles of relativity and equivalence, which postulate the equality between gravitational mass and inertial mass. To these principles are added the Hodge–Helmholtz decomposition, the principle of accumulation of constraints and the hypothesis of the duality of physical actions. These principles make it possible to establish the equation of motion based on the conservation of acceleration considered as an absolute quantity in a local frame of reference, in the form of a sum of the gradient of the scalar potential and the curl of the vector potential. These potentials, which represent the constraints of compression and rotation, are updated from the discrete operators. Discrete Mechanics: Concepts and Applications shows that this equation of discrete motion is representative of the compressible or incompressible flows of viscous or perfect fluids, the state of stress in an elastic solid or complex fluid and the propagation of nonlinear waves.

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.

Volume is indexed by Thomson Reuters CPCI-S (WoS). This collection of peer-reviewed papers describes the latest advances in, and applications of: basic mechanics and research methods, dynamics and vibration, solid mechanics, fluid mechanics and thermodynamics, biomechanics and environmental mechanics, new materials and advanced materials, functional materials, materials processing technology, welding and mechanical connections, fracture, etc. the work is thus a usefully up-to-date guide to these topics.

This textbook compiles reports written by about 35 internationally recognized authorities, and covers a range of interests for geotechnical engineers. Topics include: fundamentals for mechanics of granular materials; continuum theory of granular materials; and discrete element approaches.

After a quarter century of discoveries that rattled the foundations of classical mechanics and electrodynamics, the year 1926 saw the publication of two works intended to provide a theoretical structure to support new quantum explanations of the subatomic world. Heisenberg's matrix mechanics and Schrodinger’s wave mechanics provided compatible but mathematically disparate ways of unifying the discoveries of Planck, Einstein, Bohr and many others. Efforts began immediately to prove the equivalence of these two structures, culminated successfully by John von Neumann’s 1932 volume "Mathematical Foundations of Quantum Mechanics." This forms the springboard for the current effort. We begin with a presentation of a minimal set of von Neumann postulates while introducing language and notation to facilitate subsequent discussion of quantum calculations based in finite dimensional Hilbert spaces. Chapters which follow address two-state quantum systems (with spin one-half as the primary example), entanglement of multiple two-state systems, quantum angular momentum theory and quantum approaches to statistical mechanics. A concluding chapter gives an overview of issues associated with quantum mechanics in continuous infinite-dimensional Hilbert spaces.

This in-depth resource teaches you to craft mechanics that generate challenging, enjoyable, and well-balanced gameplay. You’ll discover at what stages to prototype, test, and implement mechanics in games and learn how to visualize and simulate game mechanics in order to design better games. Along the way, you’ll practice what you’ve learned with hands-on lessons. A free downloadable simulation tool developed by Joris Dormans is also available in order to follow along with exercises in the book in an easy-to-use graphical environment. In Game Mechanics: Advanced Game Design, you’ll learn how to: * Design and balance game mechanics to create emergent gameplay before you write a single line of code. * Visualize the internal economy so that you can immediately see what goes on in a complex game. * Use novel prototyping techniques that let you simulate games and collect vast quantities of gameplay data on the first day of development. * Apply design patterns for game mechanics—from a library in this book—to improve your game designs. * Explore the delicate balance between game mechanics and level design to create compelling, long-lasting game experiences. * Replace fixed, scripted events in your game with dynamic progression systems to give your players a new experience every time they play. "I've been waiting for a book like this for ten years: packed with game design goodness that tackles the science without undermining the art." --Richard Bartle, University of Essex, co-author of the first MMORPG “Game Mechanics: Advanced Game Design by Joris Dormans & Ernest Adams formalizes game grammar quite well. Not sure I need to write a next book now!” -- Raph Koster, author of A Theory of Fun for Game Design.

Constitutive equations define the response of materials which are subjected to applied fields. This volume presents the procedures for generating constitutive equations describing the response of crystals, isotropic and transversely isotropic materials. The book discusses the application of group representation theory, Young symmetry operators and generating functions to the determination of the general form of constitutive equations. Basic quantity tables, character tables, irreducible representation tables and direct product tables are included.

Arithmetic Applied Mathematics deals with concepts of arithmetic applied mathematics and uses a computer, rather than a continuum, approach to the deterministic theories of particle mechanics. Models of classical physical phenomena are formulated from both Newtonian and special relativistic mechanics using only arithmetic. Definitions of energy and momentum are presented that are identical to those of continuum mechanics. Comprised of nine chapters, this book begins by exploring discrete modeling as it relates to Newtonian mechanics and special relativistic mechanics, paying particular attention to gravity. The reader is then introduced to long-range forces such as gravitation and short-range forces such as molecular attraction and repulsion; the N-body problem; and conservative and non-conservative models of complex physical phenomena. Subsequent chapters focus on the foundational concepts of special relativity; arithmetic special relativistic mechanics in one space dimension and three space dimensions; and Lorentz invariant computations. This monograph will be of interest to students and practitioners in the fields of mathematics and physics.