Mathematical Methods of Classical Mechanics

Author: V.I. Arnol'd
Publisher: Springer Science & Business Media
ISBN: 9781475720631
Release Date: 2013-04-09
Genre: Mathematics

This book constructs the mathematical apparatus of classical mechanics from the beginning, examining basic problems in dynamics like the theory of oscillations and the Hamiltonian formalism. The author emphasizes geometrical considerations and includes phase spaces and flows, vector fields, and Lie groups. Discussion includes qualitative methods of the theory of dynamical systems and of asymptotic methods like averaging and adiabatic invariance.

Mathematical Methods of Classical Mechanics

Author: V.I. Arnol'd
Publisher: Springer Science & Business Media
ISBN: 0387968903
Release Date: 1997-09-05
Genre: Mathematics

This book constructs the mathematical apparatus of classical mechanics from the beginning, examining basic problems in dynamics like the theory of oscillations and the Hamiltonian formalism. The author emphasizes geometrical considerations and includes phase spaces and flows, vector fields, and Lie groups. Discussion includes qualitative methods of the theory of dynamical systems and of asymptotic methods like averaging and adiabatic invariance.

Mathematics of Classical and Quantum Physics

Author: Frederick W. Byron
Publisher: Courier Corporation
ISBN: 9780486135069
Release Date: 2012-04-26
Genre: Science

Graduate-level text offers unified treatment of mathematics applicable to many branches of physics. Theory of vector spaces, analytic function theory, theory of integral equations, group theory, and more. Many problems. Bibliography.

The Variational Principles of Mechanics

Author: Cornelius Lanczos
Publisher: Courier Corporation
ISBN: 9780486134703
Release Date: 2012-04-24
Genre: Science

Philosophic, less formalistic approach to analytical mechanics offers model of clear, scholarly exposition at graduate level with coverage of basics, calculus of variations, principle of virtual work, equations of motion, more.

Applications of Lie Groups to Differential Equations

Author: Peter J. Olver
Publisher: Springer Science & Business Media
ISBN: 9781468402742
Release Date: 2012-12-06
Genre: Mathematics

This book is devoted to explaining a wide range of applications of con tinuous symmetry groups to physically important systems of differential equations. Emphasis is placed on significant applications of group-theoretic methods, organized so that the applied reader can readily learn the basic computational techniques required for genuine physical problems. The first chapter collects together (but does not prove) those aspects of Lie group theory which are of importance to differential equations. Applications covered in the body of the book include calculation of symmetry groups of differential equations, integration of ordinary differential equations, including special techniques for Euler-Lagrange equations or Hamiltonian systems, differential invariants and construction of equations with pre scribed symmetry groups, group-invariant solutions of partial differential equations, dimensional analysis, and the connections between conservation laws and symmetry groups. Generalizations of the basic symmetry group concept, and applications to conservation laws, integrability conditions, completely integrable systems and soliton equations, and bi-Hamiltonian systems are covered in detail. The exposition is reasonably self-contained, and supplemented by numerous examples of direct physical importance, chosen from classical mechanics, fluid mechanics, elasticity and other applied areas.

Introduction to Dynamics

Author: I. C. Percival
Publisher: Cambridge University Press
ISBN: 0521281490
Release Date: 1982-12-02
Genre: Mathematics

A new approach to dynamics that takes account of recent advances that have wide applications in the sciences and engineering. It introduces the subject at an undergraduate level by means of elementary qualitative theory of differential equations, the geometry of phase curves, and the theory of stability.

Theory of Oscillations

Author: Vladimir I Zubov
Publisher: World Scientific
ISBN: 9789814518017
Release Date: 1999-02-04
Genre: Mathematics

This monograph deals with the controlled/non-controlled nonlinear systems of differential equations. A mathematical apparatus is developed to construct stationary conditions and to carry out studies on the behaviour of integral curves in the neighbourhood of such conditions. Considerable coverage is given to existence and methods of finding periodic orbits and almost-periodic solutions, as well as to the description of the class of ergodic recurrent motions. There is further treatment of the perturbation method and the theory of time-independent and periodic perturbations in particular. The theory developed here is applied to the construction and investigation of the neigbourhood of time-independent conditions for nonlinear systems of automatic control, and the control of charged particle beam in magnetic field. Some other specific problems are also solved such as after effect systems and orbit quantization. Contents:Preliminary Representations and Analyses of Motion Family BehaviorOn Behavior of Trajectories in the Neighborhood of a Periodic OrbitNatural and Forced Oscillations in Systems with Many Degrees of FreedomMethods for Investigation and Construction of Stationary ModesOscillations in Nonlinear and Controlled SystemsAppendix: Theory of Rated Stability Readership: Mathematicians and physicists. keywords:Theory of Oscillations;Behavior of Integral Curves;Ordinary Differential Equations;Autonomous Dynamical Systems;Periodic Solutions;Almost-Periodic Solutions;Recurrent Functions;Nonlinear Oscillations;Stability of Motions

Arnold s Problems

Author: Vladimir I. Arnold
Publisher: Springer Science & Business Media
ISBN: 3540206140
Release Date: 2004-06-24
Genre: Mathematics

Vladimir Arnold is one of the most outstanding mathematicians of our time Many of these problems are at the front line of current research

Geometrical Methods in the Theory of Ordinary Differential Equations

Author: V.I. Arnold
Publisher: Springer Science & Business Media
ISBN: 9781461210375
Release Date: 2012-12-06
Genre: Mathematics

Since the first edition of this book, geometrical methods in the theory of ordinary differential equations have become very popular and some progress has been made partly with the help of computers. Much of this progress is represented in this revised, expanded edition, including such topics as the Feigenbaum universality of period doubling, the Zoladec solution, the Iljashenko proof, the Ecalle and Voronin theory, the Varchenko and Hovanski theorems, and the Neistadt theory. In the selection of material for this book, the author explains basic ideas and methods applicable to the study of differential equations. Special efforts were made to keep the basic ideas free from excessive technicalities. Thus the most fundamental questions are considered in great detail, while of the more special and difficult parts of the theory have the character of a survey. Consequently, the reader needs only a general mathematical knowledge to easily follow this text. It is directed to mathematicians, as well as all users of the theory of differential equations.

Structure of Dynamical Systems

Author: J.M. Souriau
Publisher: Springer Science & Business Media
ISBN: 9781461202813
Release Date: 2012-12-06
Genre: Mathematics

The aim of the book is to treat all three basic theories of physics, namely, classical mechanics, statistical mechanics, and quantum mechanics from the same perspective, that of symplectic geometry, thus showing the unifying power of the symplectic geometric approach. Reading this book will give the reader a deep understanding of the interrelationships between the three basic theories of physics. This book is addressed to graduate students and researchers in mathematics and physics who are interested in mathematical and theoretical physics, symplectic geometry, mechanics, and (geometric) quantization.

Foundations of Mechanics

Author: Ralph Abraham
Publisher: American Mathematical Soc.
ISBN: 9780821844380
Release Date: 1978
Genre: Mathematics

Undoubtedly [the book] will be for years the standard reference on symplectic geometry, analytical mechanics and symplectic methods in mathematical physics. --Zentralblatt fur Mathematik For many years, this book has been viewed as a classic treatment of geometric mechanics. It is known for its broad exposition of the subject, with many features that cannot be found elsewhere. The book is recommended as a textbook and as a basic reference work for the foundations of differentiable and Hamiltonian dynamics.

Theoretical Mechanics of Particles and Continua

Author: Alexander L. Fetter
Publisher: Courier Corporation
ISBN: 9780486432618
Release Date: 2003-12-16
Genre: Science

This two-part text fills what has often been a void in the first-year graduate physics curriculum. Through its examination of particles and continua, it supplies a lucid and self-contained account of classical mechanics — which in turn provides a natural framework for introducing many of the advanced mathematical concepts in physics. The text opens with Newton's laws of motion and systematically develops the dynamics of classical particles, with chapters on basic principles, rotating coordinate systems, lagrangian formalism, small oscillations, dynamics of rigid bodies, and hamiltonian formalism, including a brief discussion of the transition to quantum mechanics. This part of the book also considers examples of the limiting behavior of many particles, facilitating the eventual transition to a continuous medium. The second part deals with classical continua, including chapters on string membranes, sound waves, surface waves on nonviscous fluids, heat conduction, viscous fluids, and elastic media. Each of these self-contained chapters provides the relevant physical background and develops the appropriate mathematical techniques, and problems of varying difficulty appear throughout the text.

Quantum Theory for Mathematicians

Author: Brian C. Hall
Publisher: Springer Science & Business Media
ISBN: 9781461471165
Release Date: 2013-06-19
Genre: Science

Although ideas from quantum physics play an important role in many parts of modern mathematics, there are few books about quantum mechanics aimed at mathematicians. This book introduces the main ideas of quantum mechanics in language familiar to mathematicians. Readers with little prior exposure to physics will enjoy the book's conversational tone as they delve into such topics as the Hilbert space approach to quantum theory; the Schrödinger equation in one space dimension; the Spectral Theorem for bounded and unbounded self-adjoint operators; the Stone–von Neumann Theorem; the Wentzel–Kramers–Brillouin approximation; the role of Lie groups and Lie algebras in quantum mechanics; and the path-integral approach to quantum mechanics. The numerous exercises at the end of each chapter make the book suitable for both graduate courses and independent study. Most of the text is accessible to graduate students in mathematics who have had a first course in real analysis, covering the basics of L2 spaces and Hilbert spaces. The final chapters introduce readers who are familiar with the theory of manifolds to more advanced topics, including geometric quantization.

Mathematics for Physics

Author: Michael Stone
Publisher: Cambridge University Press
ISBN: 9781139480611
Release Date: 2009-07-09
Genre: Science

An engagingly-written account of mathematical tools and ideas, this book provides a graduate-level introduction to the mathematics used in research in physics. The first half of the book focuses on the traditional mathematical methods of physics – differential and integral equations, Fourier series and the calculus of variations. The second half contains an introduction to more advanced subjects, including differential geometry, topology and complex variables. The authors' exposition avoids excess rigor whilst explaining subtle but important points often glossed over in more elementary texts. The topics are illustrated at every stage by carefully chosen examples, exercises and problems drawn from realistic physics settings. These make it useful both as a textbook in advanced courses and for self-study. Password-protected solutions to the exercises are available to instructors at www.cambridge.org/9780521854030.