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This volume in the Encyclopedia of Complexity and Systems Science, Second Edition, is devoted to the fundamentals of Perturbation Theory (PT) as well as key applications areas such as Classical and Quantum Mechanics, Celestial Mechanics, and Molecular Dynamics. Less traditional fields of application, such as Biological Evolution, are also discussed. Leading scientists in each area of the field provide a comprehensive picture of the landscape and the state of the art, with the specific goal of combining mathematical rigor, explicit computational methods, and relevance to concrete applications. New to this edition are chapters on Water Waves, Rogue Waves, Multiple Scales methods, legged locomotion, Condensed Matter among others, while all other contributions have been revised and updated. Coverage includes the theory of (Poincare’-Birkhoff) Normal Forms, aspects of PT in specific mathematical settings (Hamiltonian, KAM theory, Nekhoroshev theory, and symmetric systems), technical problems arising in PT with solutions, convergence of series expansions, diagrammatic methods, parametric resonance, systems with nilpotent real part, PT for non-smooth systems, and on PT for PDEs [write out this acronym partial differential equations]. Another group of papers is focused specifically on applications to Celestial Mechanics, Quantum Mechanics and the related semiclassical PT, Quantum Bifurcations, Molecular Dynamics, the so-called choreographies in the N-body problem, as well as Evolutionary Theory. Overall, this unique volume serves to demonstrate the wide utility of PT, while creating a foundation for innovations from a new generation of graduate students and professionals in Physics, Mathematics, Mechanics, Engineering and the Biological Sciences.

Perturbation (Mathematics) --- Perturbation equations --- Perturbation theory --- Approximation theory --- Dynamics --- Functional analysis --- Mathematical physics --- Mathematical physics. --- Civil engineering. --- Mechanics. --- Evolution (Biology). --- Theoretical, Mathematical and Computational Physics. --- Mathematical Physics. --- Civil Engineering. --- Classical Mechanics. --- Evolutionary Biology. --- Engineering --- Public works --- Animal evolution --- Animals --- Biological evolution --- Darwinism --- Evolutionary biology --- Evolutionary science --- Origin of species --- Biology --- Evolution --- Biological fitness --- Homoplasy --- Natural selection --- Phylogeny --- Classical mechanics --- Newtonian mechanics --- Physics --- Quantum theory --- Physical mathematics --- Mathematics

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This volume in the Encyclopedia of Complexity and Systems Science, Second Edition, is devoted to the fundamentals of Perturbation Theory (PT) as well as key applications areas such as Classical and Quantum Mechanics, Celestial Mechanics, and Molecular Dynamics. Less traditional fields of application, such as Biological Evolution, are also discussed. Leading scientists in each area of the field provide a comprehensive picture of the landscape and the state of the art, with the specific goal of combining mathematical rigor, explicit computational methods, and relevance to concrete applications. New to this edition are chapters on Water Waves, Rogue Waves, Multiple Scales methods, legged locomotion, Condensed Matter among others, while all other contributions have been revised and updated. Coverage includes the theory of (Poincare'-Birkhoff) Normal Forms, aspects of PT in specific mathematical settings (Hamiltonian, KAM theory, Nekhoroshev theory, and symmetric systems), technical problems arising in PT with solutions, convergence of series expansions, diagrammatic methods, parametric resonance, systems with nilpotent real part, PT for non-smooth systems, and on PT for PDEs [write out this acronym partial differential equations]. Another group of papers is focused specifically on applications to Celestial Mechanics, Quantum Mechanics and the related semiclassical PT, Quantum Bifurcations, Molecular Dynamics, the so-called choreographies in the N-body problem, as well as Evolutionary Theory. Overall, this unique volume serves to demonstrate the wide utility of PT, while creating a foundation for innovations from a new generation of graduate students and professionals in Physics, Mathematics, Mechanics, Engineering and the Biological Sciences.

Mathematical physics --- Classical mechanics. Field theory --- Evolution. Phylogeny --- Civil engineering. Building industry --- theoretische fysica --- Europees recht --- evolutieleer --- wiskunde --- ingenieurswetenschappen --- fysica --- mechanica

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This book provides the mathematical foundations of the theory of hyperhamiltonian dynamics, together with a discussion of physical applications. In addition, some open problems are discussed. Hyperhamiltonian mechanics represents a generalization of Hamiltonian mechanics, in which the role of the symplectic structure is taken by a hyperkähler one (thus there are three Kähler/symplectic forms satisfying quaternionic relations). This has proved to be of use in the description of physical systems with spin, including those which do not admit a Hamiltonian formulation. The book is the first monograph on the subject, which has previously been treated only in research papers.

Physics. --- Mathematical physics. --- Quantum physics. --- Mathematical Methods in Physics. --- Applied and Technical Physics. --- Quantum Physics. --- Mathematical Physics. --- Classical Mechanics. --- Hamiltonian systems. --- Hamiltonian dynamical systems --- Systems, Hamiltonian --- Differentiable dynamical systems --- Quantum theory. --- Mechanics. --- Classical mechanics --- Newtonian mechanics --- Physics --- Dynamics --- Quantum theory --- Quantum dynamics --- Quantum mechanics --- Quantum physics --- Mechanics --- Thermodynamics --- Physical mathematics --- Mathematics --- Natural philosophy --- Philosophy, Natural --- Physical sciences

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This book provides the mathematical foundations of the theory of hyperhamiltonian dynamics, together with a discussion of physical applications. In addition, some open problems are discussed. Hyperhamiltonian mechanics represents a generalization of Hamiltonian mechanics, in which the role of the symplectic structure is taken by a hyperkähler one (thus there are three Kähler/symplectic forms satisfying quaternionic relations). This has proved to be of use in the description of physical systems with spin, including those which do not admit a Hamiltonian formulation. The book is the first monograph on the subject, which has previously been treated only in research papers.

Mathematical physics --- Quantum mechanics. Quantumfield theory --- Classical mechanics. Field theory --- Physics --- Applied physical engineering --- quantumfysica --- toegepaste wetenschappen --- quantumtheorie --- wiskunde --- ingenieurswetenschappen --- fysica --- mechanica

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The third conference on "Symmetry and Perturbation Theory" (SPT2001) was attended by over 50 mathematicians, physicists and chemists. The proceedings present the advancement of research in this field - more precisely, in the different fields at whose crossroads symmetry and perturbation theory sit.
Contents:

• Geometry and Dynamics of Hyperelliptically Separable Systems (S Abenda)
• A Functional Analysis Approach to Arnold Diffusion (M Berti)
• Linearizing Resonant Normal Forms (G Gaeta)
• Tori Breakdown in Coupled Map Lattices (C Gibe