Nonlinear Oscillations of Hamiltonian PDEs
Nonlinear Oscillations of Hamiltonian PDEs

Author: Massimiliano Berti

Publisher: Springer Science & Business Media

Published: 2007-10-01

Total Pages: 191

ISBN-13: 0817646809

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Many partial differential equations (PDEs) that arise in physics can be viewed as infinite-dimensional Hamiltonian systems. This monograph presents recent existence results of nonlinear oscillations of Hamiltonian PDEs, particularly of periodic solutions for completely resonant nonlinear wave equations. The text serves as an introduction to research in this fascinating and rapidly growing field. Graduate students and researchers interested in variational techniques and nonlinear analysis applied to Hamiltonian PDEs will find inspiration in the book.

Nonlinear Oscillations of Hamiltonian PDEs
Nonlinear Oscillations of Hamiltonian PDEs

Author: Massimiliano Berti

Publisher: Springer Science & Business Media

Published: 2007-10-05

Total Pages: 191

ISBN-13: 0817646817

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Many partial differential equations (PDEs) that arise in physics can be viewed as infinite-dimensional Hamiltonian systems. This monograph presents recent existence results of nonlinear oscillations of Hamiltonian PDEs, particularly of periodic solutions for completely resonant nonlinear wave equations. The text serves as an introduction to research in this fascinating and rapidly growing field. Graduate students and researchers interested in variational techniques and nonlinear analysis applied to Hamiltonian PDEs will find inspiration in the book.

Nonlinear Oscillations and Waves in Dynamical Systems
Nonlinear Oscillations and Waves in Dynamical Systems

Author: P.S Landa

Publisher: Springer Science & Business Media

Published: 2013-06-29

Total Pages: 550

ISBN-13: 9401587639

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A rich variety of books devoted to dynamical chaos, solitons, self-organization has appeared in recent years. These problems were all considered independently of one another. Therefore many of readers of these books do not suspect that the problems discussed are divisions of a great generalizing science - the theory of oscillations and waves. This science is not some branch of physics or mechanics, it is a science in its own right. It is in some sense a meta-science. In this respect the theory of oscillations and waves is closest to mathematics. In this book we call the reader's attention to the present-day theory of non-linear oscillations and waves. Oscillatory and wave processes in the systems of diversified physical natures, both periodic and chaotic, are considered from a unified poin t of view . The relation between the theory of oscillations and waves, non-linear dynamics and synergetics is discussed. One of the purposes of this book is to convince reader of the necessity of a thorough study popular branches of of the theory of oscillat ions and waves, and to show that such science as non-linear dynamics, synergetics, soliton theory, and so on, are, in fact , constituent parts of this theory. The primary audiences for this book are researchers having to do with oscillatory and wave processes, and both students and post-graduate students interested in a deep study of the general laws and applications of the theory of oscillations and waves.

Hamiltonian Dynamical Systems and Applications
Hamiltonian Dynamical Systems and Applications

Author: Walter Craig

Publisher: Springer Science & Business Media

Published: 2008-02-17

Total Pages: 450

ISBN-13: 1402069642

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This volume is the collected and extended notes from the lectures on Hamiltonian dynamical systems and their applications that were given at the NATO Advanced Study Institute in Montreal in 2007. Many aspects of the modern theory of the subject were covered at this event, including low dimensional problems. Applications are also presented to several important areas of research, including problems in classical mechanics, continuum mechanics, and partial differential equations.

Numerical Continuation and Bifurcation in Nonlinear PDEs
Numerical Continuation and Bifurcation in Nonlinear PDEs

Author: Hannes Uecker

Publisher: SIAM

Published: 2021-08-19

Total Pages: 380

ISBN-13: 1611976618

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This book provides a hands-on approach to numerical continuation and bifurcation for nonlinear PDEs in 1D, 2D, and 3D. Partial differential equations (PDEs) are the main tool to describe spatially and temporally extended systems in nature. PDEs usually come with parameters, and the study of the parameter dependence of their solutions is an important task. Letting one parameter vary typically yields a branch of solutions, and at special parameter values, new branches may bifurcate. After a concise review of some analytical background and numerical methods, the author explains the free MATLAB package pde2path by using a large variety of examples with demo codes that can be easily adapted to the reader's given problem. Numerical Continuation and Bifurcation in Nonlinear PDEs will appeal to applied mathematicians and scientists from physics, chemistry, biology, and economics interested in the numerical solution of nonlinear PDEs, particularly the parameter dependence of solutions. It can be used as a supplemental text in courses on nonlinear PDEs and modeling and bifurcation.

Nonlinear Equations for Beams and Degenerate Plates with Piers
Nonlinear Equations for Beams and Degenerate Plates with Piers

Author: Maurizio Garrione

Publisher: Springer Nature

Published: 2019-10-31

Total Pages: 115

ISBN-13: 3030302180

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This book develops a full theory for hinged beams and degenerate plates with multiple intermediate piers with the final purpose of understanding the stability of suspension bridges. New models are proposed and new tools are provided for the stability analysis. The book opens by deriving the PDE’s based on the physical models and by introducing the basic framework for the linear stationary problem. The linear analysis, in particular the behavior of the eigenvalues as the position of the piers varies, enables the authors to tackle the stability issue for some nonlinear evolution beam equations, with the aim of determining the “best position” of the piers within the beam in order to maximize its stability. The study continues with the analysis of a class of degenerate plate models. The torsional instability of the structure is investigated, and again, the optimal position of the piers in terms of stability is discussed. The stability analysis is carried out by means of both analytical tools and numerical experiments. Several open problems and possible future developments are presented. The qualitative analysis provided in the book should be seen as the starting point for a precise quantitative study of more complete models, taking into account the action of aerodynamic forces. This book is intended for a two-fold audience. It is addressed both to mathematicians working in the field of Differential Equations, Nonlinear Analysis and Mathematical Physics, due to the rich number of challenging mathematical questions which are discussed and left as open problems, and to Engineers interested in mechanical structures, since it provides the theoretical basis to deal with models for the dynamics of suspension bridges with intermediate piers. More generally, it may be enjoyable for readers who are interested in the application of Mathematics to real life problems.

Lectures On Quantum Mechanics And Attractors
Lectures On Quantum Mechanics And Attractors

Author: Alexander Komech

Publisher: World Scientific

Published: 2022-02-18

Total Pages: 272

ISBN-13: 9811248915

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This book gives a concise introduction to Quantum Mechanics with a systematic, coherent, and in-depth explanation of related mathematical methods from the scattering theory and the theory of Partial Differential Equations.The book is aimed at graduate and advanced undergraduate students in mathematics, physics, and chemistry, as well as at the readers specializing in quantum mechanics, theoretical physics and quantum chemistry, and applications to solid state physics, optics, superconductivity, and quantum and high-frequency electronic devices.The book utilizes elementary mathematical derivations. The presentation assumes only basic knowledge of the origin of Hamiltonian mechanics, Maxwell equations, calculus, Ordinary Differential Equations and basic PDEs. Key topics include the Schrödinger, Pauli, and Dirac equations, the corresponding conservation laws, spin, the hydrogen spectrum, and the Zeeman effect, scattering of light and particles, photoelectric effect, electron diffraction, and relations of quantum postulates with attractors of nonlinear Hamiltonian PDEs. Featuring problem sets and accompanied by extensive contemporary and historical references, this book could be used for the course on Quantum Mechanics and is also suitable for individual study.

Variational Methods
Variational Methods

Author: Maïtine Bergounioux

Publisher: Walter de Gruyter GmbH & Co KG

Published: 2017-01-11

Total Pages: 621

ISBN-13: 3110430495

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With a focus on the interplay between mathematics and applications of imaging, the first part covers topics from optimization, inverse problems and shape spaces to computer vision and computational anatomy. The second part is geared towards geometric control and related topics, including Riemannian geometry, celestial mechanics and quantum control. Contents: Part I Second-order decomposition model for image processing: numerical experimentation Optimizing spatial and tonal data for PDE-based inpainting Image registration using phase・amplitude separation Rotation invariance in exemplar-based image inpainting Convective regularization for optical flow A variational method for quantitative photoacoustic tomography with piecewise constant coefficients On optical flow models for variational motion estimation Bilevel approaches for learning of variational imaging models Part II Non-degenerate forms of the generalized Euler・Lagrange condition for state-constrained optimal control problems The Purcell three-link swimmer: some geometric and numerical aspects related to periodic optimal controls Controllability of Keplerian motion with low-thrust control systems Higher variational equation techniques for the integrability of homogeneous potentials Introduction to KAM theory with a view to celestial mechanics Invariants of contact sub-pseudo-Riemannian structures and Einstein・Weyl geometry Time-optimal control for a perturbed Brockett integrator Twist maps and Arnold diffusion for diffeomorphisms A Hamiltonian approach to sufficiency in optimal control with minimal regularity conditions: Part I Index

Advanced Topics in Nonsmooth Dynamics
Advanced Topics in Nonsmooth Dynamics

Author: Remco Leine

Publisher: Springer

Published: 2018-06-07

Total Pages: 462

ISBN-13: 3319759728

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This book discusses emerging topics in the area of nonsmooth dynamics research, such as numerical methods for nonsmooth systems, impact laws for multi-collisions, nonlinear vibrations and control of nonsmooth systems. It documents original work of researchers at the European Network for NonSmooth Dynamics (ENNSD), which provides a cooperation platform for researchers in the field and promotes research focused on nonsmooth dynamics and its applications. Since the establishment of the network in 2012, six ENNSD symposia have been organized at different European locations. The network brings together 40 specialists from 9 different countries in and outside Europe and a wealth of scientific knowledge has been gathered and developed by this group of experts in recent years. The book is of interest to both new and experienced researchers in the field of nonsmooth dynamics. Each chapter is written in such a way as to provide an introduction to the topic for researchers from other fields.

The Parameterization Method for Invariant Manifolds
The Parameterization Method for Invariant Manifolds

Author: Àlex Haro

Publisher: Springer

Published: 2016-04-18

Total Pages: 280

ISBN-13: 3319296620

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This monograph presents some theoretical and computational aspects of the parameterization method for invariant manifolds, focusing on the following contexts: invariant manifolds associated with fixed points, invariant tori in quasi-periodically forced systems, invariant tori in Hamiltonian systems and normally hyperbolic invariant manifolds. This book provides algorithms of computation and some practical details of their implementation. The methodology is illustrated with 12 detailed examples, many of them well known in the literature of numerical computation in dynamical systems. A public version of the software used for some of the examples is available online. The book is aimed at mathematicians, scientists and engineers interested in the theory and applications of computational dynamical systems.