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As a limit theory of quantum mechanics, classical dynamics comprises a large variety of phenomena, from computable (integrable) to chaotic (mixing) behavior. This book presents the KAM (Kolmogorov-Arnold-Moser) theory and asymptotic completeness in classical scattering. Including a wealth of fascinating examples in physics, it offers not only an excellent selection of basic topics, but also an introduction to a number of current areas of research in the field of classical mechanics. Thanks to the didactic structure and concise appendices, the presentation is self-contained and requires only knowledge of the basic courses in mathematics. The book addresses the needs of graduate and senior undergraduate students in mathematics and physics, and of researchers interested in approaching classical mechanics from a modern point of view.

Mathematics. --- Mathematical physics. --- Physics. --- Mathematical Physics. --- Theoretical, Mathematical and Computational Physics. --- Physical mathematics --- Physics --- Mathematics

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By systematically covering both the Heisenberg and Schrödinger realizations, the book emphasizes the essential principles of quantum mechanics, which remain hidden within the usual derivations of the wave equation. Moreover, this presentation not only covers the material which is traditionally presented in textbooks, but also is especially suitable for introducing the spin, i.e., the most important quantum observable. This emphasis on spin paves the way for a presentation of recent quantum-mechanical concepts such as entanglement and decoherence, and to recent applications including cryptography, teleportation and quantum computation. "I am very impressed with Dr. Bes' approach to the subject, the clarity of his exposition, and the timeliness of the examples, many of which are taken from the most recent developments of the "old-new" field of quantum mechanics" (Prof. J. Roederer)

Quantum theory --- Quantum theory. --- Quantum Physics. --- Theoretical, Mathematical and Computational Physics. --- Quantum dynamics --- Quantum mechanics --- Quantum physics --- Physics --- Mechanics --- Thermodynamics --- Quantum physics. --- Mathematical physics. --- Physical mathematics --- Mathematics

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This book presents the basic elements of Analytical Mechanics, starting from the physical motivations that favor it with respect to the Newtonian Mechanics in Cartesian coordinates. Rather than presenting Analytical Mechanics mainly as a formal development of Newtonian Mechanics, it highlights its effectiveness due to the following five important achievements: 1) the most economical description of time evolution in terms of the minimal set of coordinates, so that there are no constraint forces in their evolution equations; 2) the form invariance of the evolution equations, which automatically solves the problem of fictitious forces; 3) only one scalar function encodes the formulation of the dynamics, rather than the full set of vectors which describe the forces in Cartesian Newtonian Mechanics; 4) in the Hamiltonian formulation, the corresponding evolution equations are of first order in time and are fully governed by the Hamiltonian function (usually corresponding to the energy); 5) the emergence of the Hamiltonian canonical algebra and its effectiveness in simplifying the control of the dynamical problem (e.g. the constant of motions identified by the Poisson brackets with the Hamiltonian, the relation between symmetries and conservations laws, the use of canonical transformations to reduce the Hamiltonian to a simpler form etc.). The book also addresses a number of points usually not included in textbook presentations of Analytical Mechanics, such as 1) the characterization of the cases in which the Hamiltonian differs from the energy, 2) the characterization of the non-uniqueness of the Lagrangian and of the Hamiltonian and its relation to a “gauge” transformation, 3) the Hamiltonian formulation of the Noether theorem, with the possibility that the constant of motion corresponding to a continuous symmetry of the dynamics is not the canonical generator of the symmetry transformation but also involves the generator of a gauge transformation. In turn, the book’s closing chapter is devoted to explaining the extraordinary analogy between the canonical structure of Classical and Quantum Mechanics. By correcting the Dirac proposal for such an explanation, it demonstrates that there is a common Poisson algebra shared by Classical and Quantum Mechanics, the differences between the two theories being reducible to the value of the central variable of that algebra.

Physics. --- Mathematical physics. --- Classical Mechanics. --- Mathematical Physics. --- Theoretical, Mathematical and Computational Physics. --- Mechanics. --- Classical mechanics --- Newtonian mechanics --- Physics --- Dynamics --- Quantum theory --- Physical mathematics --- Mathematics

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Based on 3D smoothed particle hydrodynamics simulations performed with unprecedented high resolution, this book examines the giant impacts that dominate many planets’ late accretion and evolution. The numerical methods developed are now publicly available, greatly facilitating future studies of planetary impacts in our solar system and exoplanetary systems. The book focuses on four main topics: (1) The development of new methods to construct initial conditions as well as a hydrodynamical simulation code to evolve them, using 1000 times more simulation particles than the previous standard. (2) The numerical convergence of giant impact simulations -- standard-resolution simulations fail to converge on even bulk properties like the post-impact rotation period. (3) The collision thought to have knocked over the planet Uranus causing it to spin on its side. (4) The erosion of atmospheres by giant impacts onto terrestrial planets, and the first full 3D simulations of collisions in this regime.

Astronomy. --- Astrophysics. --- Mathematical physics. --- Astronomy, Astrophysics and Cosmology. --- Astrophysics and Astroparticles. --- Theoretical, Mathematical and Computational Physics. --- Physical mathematics --- Physics --- Astronomical physics --- Astronomy --- Cosmic physics --- Mathematics

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This monograph is devoted to the study of multiscale model reduction methods from the point of view of multiscale finite element methods. Multiscale numerical methods have become popular tools for modeling processes with multiple scales. These methods allow reducing the degrees of freedom based on local offline computations. Moreover, these methods allow deriving rigorous macroscopic equations for multiscale problems without scale separation and high contrast. Multiscale methods are also used to design efficient solvers. This book offers a combination of analytical and numerical methods designed for solving multiscale problems. The book mostly focuses on methods that are based on multiscale finite element methods. Both applications and theoretical developments in this field are presented. The book is suitable for graduate students and researchers, who are interested in this topic.

Numerical analysis. --- Mathematics—Data processing. --- Mathematical physics. --- Numerical Analysis. --- Computational Science and Engineering. --- Theoretical, Mathematical and Computational Physics. --- Physical mathematics --- Physics --- Mathematics --- Mathematical analysis --- Modelització multiescala