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Mathematical Methods of Analytical Mechanics

Mathematical Methods of Analytical Mechanics
A Book

by Henri Gouin

  • Publisher : Elsevier
  • Release : 2020-11-27
  • Pages : 320
  • ISBN : 0128229861
  • Language : En, Es, Fr & De
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Mathematical Methods of Analytical Mechanics uses tensor geometry and geometry of variation calculation, includes the properties associated with Noether's theorem, and highlights methods of integration, including Jacobi's method, which is deduced. In addition, the book covers the Maupertuis principle that looks at the conservation of energy of material systems and how it leads to quantum mechanics. Finally, the book deduces the various spaces underlying the analytical mechanics which lead to the Poisson algebra and the symplectic geometry. Helps readers understand calculations surrounding the geometry of the tensor and the geometry of the calculation of the variation Presents principles that correspond to the energy conservation of material systems Defines the invariance properties associated with Noether's theorem Discusses phase space and Liouville's theorem Identifies small movements and different types of stabilities

Mathematical Methods of Classical Mechanics

Mathematical Methods of Classical Mechanics
A Book

by V. I. Arnold

  • Publisher : Springer Science & Business Media
  • Release : 2013-11-11
  • Pages : 464
  • ISBN : 1475716931
  • Language : En, Es, Fr & De
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Many different mathematical methods and concepts are used in classical mechanics: differential equations and phase ftows, smooth mappings and manifolds, Lie groups and Lie algebras, symplectic geometry and ergodic theory. Many modern mathematical theories arose from problems in mechanics and only later acquired that axiomatic-abstract form which makes them so hard to study. In this book we construct the mathematical apparatus of classical mechanics from the very beginning; thus, the reader is not assumed to have any previous knowledge beyond standard courses in analysis (differential and integral calculus, differential equations), geometry (vector spaces, vectors) and linear algebra (linear operators, quadratic forms). With the help of this apparatus, we examine all the basic problems in dynamics, including the theory of oscillations, the theory of rigid body motion, and the hamiltonian formalism. The author has tried to show the geometric, qualitative aspect of phenomena. In this respect the book is closer to courses in theoretical mechanics for theoretical physicists than to traditional courses in theoretical mechanics as taught by mathematicians.

Mathematical Methods of Classical Mechanics

Mathematical Methods of Classical Mechanics
A Book

by V.I. Arnol'd

  • Publisher : Springer
  • Release : 1997-09-05
  • Pages : 520
  • ISBN : 9780387968902
  • Language : En, Es, Fr & De
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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

Mathematical Methods of Classical Mechanics
A Book

by V.I. Arnol'd

  • Publisher : Springer
  • Release : 1997-09-05
  • Pages : 520
  • ISBN : 9780387968902
  • Language : En, Es, Fr & De
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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

Mathematical Methods of Classical Mechanics
A Book

by V. I. Arnold

  • Publisher : Springer
  • Release : 1984-07-30
  • Pages : 464
  • ISBN : 9780387903149
  • Language : En, Es, Fr & De
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Many different mathematical methods and concepts are used in classical mechanics: differential equations and phase ftows, smooth mappings and manifolds, Lie groups and Lie algebras, symplectic geometry and ergodic theory. Many modern mathematical theories arose from problems in mechanics and only later acquired that axiomatic-abstract form which makes them so hard to study. In this book we construct the mathematical apparatus of classical mechanics from the very beginning; thus, the reader is not assumed to have any previous knowledge beyond standard courses in analysis (differential and integral calculus, differential equations), geometry (vector spaces, vectors) and linear algebra (linear operators, quadratic forms). With the help of this apparatus, we examine all the basic problems in dynamics, including the theory of oscillations, the theory of rigid body motion, and the hamiltonian formalism. The author has tried to show the geometric, qualitative aspect of phenomena. In this respect the book is closer to courses in theoretical mechanics for theoretical physicists than to traditional courses in theoretical mechanics as taught by mathematicians.

Modern Methods of Analytical Mechanics and their Applications

Modern Methods of Analytical Mechanics and their Applications
A Book

by Valentin V. Rumyantsev,Alexander V. Karapetyan

  • Publisher : Springer
  • Release : 2014-05-04
  • Pages : 344
  • ISBN : 3709125200
  • Language : En, Es, Fr & De
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The volume aims at giving a comprehensive and up-to-date view of modern methods of analytical mechanics (general equations, invariant objects, stability and bifurcations) and their applications (rigid body dynamics, celestial mechanics, multibody systems etc.). The course is at an advanced level. It is designed for postgraduate students, research engineers and academics that are familiar with basic concepts of analytical dynamics and stability theory. Although the course deals with mechanical problems, most of the concepts and methods involved are equally applicated to general dynamical systems.

Mechanical Systems, Classical Models

Mechanical Systems, Classical Models
Volume 3: Analytical Mechanics

by Petre P. Teodorescu

  • Publisher : Springer
  • Release : 2009-08-04
  • Pages : 772
  • ISBN : 9789048127634
  • Language : En, Es, Fr & De
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All phenomena in nature are characterized by motion. Mechanics deals with the objective laws of mechanical motion of bodies, the simplest form of motion. In the study of a science of nature, mathematics plays an important rôle. Mechanics is the first science of nature which has been expressed in terms of mathematics, by considering various mathematical models, associated to phenomena of the surrounding nature. Thus, its development was influenced by the use of a strong mathematical tool. As it was already seen in the first two volumes of the present book, its guideline is precisely the mathematical model of mechanics. The classical models which we refer to are in fact models based on the Newtonian model of mechanics, that is on its five principles, i.e.: the inertia, the forces action, the action and reaction, the independence of the forces action and the initial conditions principle, respectively. Other models, e.g., the model of attraction forces between the particles of a discrete mechanical system, are part of the considered Newtonian model. Kepler’s laws brilliantly verify this model in case of velocities much smaller then the light velocity in vacuum.

Methods of Differential Geometry in Analytical Mechanics

Methods of Differential Geometry in Analytical Mechanics
A Book

by M. de León,P.R. Rodrigues

  • Publisher : Elsevier
  • Release : 2011-08-18
  • Pages : 482
  • ISBN : 9780080872698
  • Language : En, Es, Fr & De
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The differential geometric formulation of analytical mechanics not only offers a new insight into Mechanics, but also provides a more rigorous formulation of its physical content from a mathematical viewpoint. Topics covered in this volume include differential forms, the differential geometry of tangent and cotangent bundles, almost tangent geometry, symplectic and pre-symplectic Lagrangian and Hamiltonian formalisms, tensors and connections on manifolds, and geometrical aspects of variational and constraint theories. The book may be considered as a self-contained text and only presupposes that readers are acquainted with linear and multilinear algebra as well as advanced calculus.

Mechanical Systems, Classical Models

Mechanical Systems, Classical Models
Volume II: Mechanics of Discrete and Continuous Systems

by Petre P. Teodorescu

  • Publisher : Springer
  • Release : 2010-11-19
  • Pages : 564
  • ISBN : 9789048180448
  • Language : En, Es, Fr & De
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As it was already seen in the first volume of the present book, its guideline is precisely the mathematical model of mechanics. The classical models which we refer to are in fact models based on the Newtonian model of mechanics, on its five principles, i. e. : the inertia, the forces action, the action and reaction, the parallelogram and the initial conditions principle, respectively. Other models, e. g. , the model of attraction forces between the particles of a discrete mechanical system, are part of the considered Newtonian model. Kepler’s laws brilliantly verify this model in case of velocities much smaller than the light velocity in vacuum. The non-classical models are relativistic and quantic. Mechanics has as object of study mechanical systems. The first volume of this book dealt with particle dynamics. The present one deals with discrete mechanical systems for particles in a number greater than the unity, as well as with continuous mechanical systems. We put in evidence the difference between these models, as well as the specificity of the corresponding studies; the generality of the proofs and of the corresponding computations yields a common form of the obtained mechanical results for both discrete and continuous systems. We mention the thoroughness by which the dynamics of the rigid solid with a fixed point has been presented. The discrete or continuous mechanical systems can be non-deformable (e. g.

Fundamental Principles of Classical Mechanics

Fundamental Principles of Classical Mechanics
A Geometrical Perspective

by Kai S Lam

  • Publisher : World Scientific Publishing Company
  • Release : 2014-07-07
  • Pages : 592
  • ISBN : 9814551503
  • Language : En, Es, Fr & De
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This book is written with the belief that classical mechanics, as a theoretical discipline, possesses an inherent beauty, depth, and richness that far transcends its immediate applications in mechanical systems. These properties are manifested, by and large, through the coherence and elegance of the mathematical structure underlying the discipline, and are eminently worthy of being communicated to physics students at the earliest stage possible. This volume is therefore addressed mainly to advanced undergraduate and beginning graduate physics students who are interested in the application of modern mathematical methods in classical mechanics, in particular, those derived from the fields of topology and differential geometry, and also to the occasional mathematics student who is interested in important physics applications of these areas of mathematics. Its main purpose is to offer an introductory and broad glimpse of the majestic edifice of the mathematical theory of classical dynamics, not only in the time-honored analytical tradition of Newton, Laplace, Lagrange, Hamilton, Jacobi, and Whittaker, but also the more topological/geometrical one established by Poincare, and enriched by Birkhoff, Lyapunov, Smale, Siegel, Kolmogorov, Arnold, and Moser (as well as many others).

Mathematical Methods for Physical and Analytical Chemistry

Mathematical Methods for Physical and Analytical Chemistry
A Book

by David Z. Goodson

  • Publisher : John Wiley & Sons
  • Release : 2011-11-14
  • Pages : 408
  • ISBN : 1118135172
  • Language : En, Es, Fr & De
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Mathematical Methods for Physical and Analytical Chemistry presents mathematical and statistical methods to students of chemistry at the intermediate, post-calculus level. The content includes a review of general calculus; a review of numerical techniques often omitted from calculus courses, such as cubic splines and Newton’s method; a detailed treatment of statistical methods for experimental data analysis; complex numbers; extrapolation; linear algebra; and differential equations. With numerous example problems and helpful anecdotes, this text gives chemistry students the mathematical knowledge they need to understand the analytical and physical chemistry professional literature.

Fundamental Principles of Classical Mechanics

Fundamental Principles of Classical Mechanics
A Geometrical Perspective

by Kai Shue Lam

  • Publisher : World Scientific Publishing Company Incorporated
  • Release : 2014
  • Pages : 574
  • ISBN : 9789814551489
  • Language : En, Es, Fr & De
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This book is written with the belief that classical mechanics, as a theoretical discipline, possesses an inherent beauty, depth, and richness that far transcends its immediate applications in mechanical systems. These properties are manifested, by and large, through the coherence and elegance of the mathematical structure underlying the discipline, and are eminently worthy of being communicated to physics students at the earliest stage possible. This volume is therefore addressed mainly to advanced undergraduate and beginning graduate physics students who are interested in the application of modern mathematical methods in classical mechanics, in particular, those derived from the fields of topology and differential geometry, and also to the occasional mathematics student who is interested in important physics applications of these areas of mathematics. Its main purpose is to offer an introductory and broad glimpse of the majestic edifice of the mathematical theory of classical dynamics, not only in the time-honored analytical tradition of Newton, Laplace, Lagrange, Hamilton, Jacobi, and Whittaker, but also the more topological/geometrical one established by Poincare, and enriched by Birkhoff, Lyapunov, Smale, Siegel, Kolmogorov, Arnold, and Moser (as well as many others).

Mathematical Aspects of Classical and Celestial Mechanics

Mathematical Aspects of Classical and Celestial Mechanics
A Book

by Vladimir I. Arnold,Valery V. Kozlov,Anatoly I. Neishtadt

  • Publisher : Springer Science & Business Media
  • Release : 2007-07-05
  • Pages : 505
  • ISBN : 3540489266
  • Language : En, Es, Fr & De
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The main purpose of the book is to acquaint mathematicians, physicists and engineers with classical mechanics as a whole, in both its traditional and its contemporary aspects. As such, it describes the fundamental principles, problems, and methods of classical mechanics, with the emphasis firmly laid on the working apparatus, rather than the physical foundations or applications. Chapters cover the n-body problem, symmetry groups of mechanical systems and the corresponding conservation laws, the problem of the integrability of the equations of motion, the theory of oscillations and perturbation theory.

Modern Methods of Analytical Mechanics and their Applications

Modern Methods of Analytical Mechanics and their Applications
A Book

by Valentin V. Rumyantsev,Alexander V. Karapetyan

  • Publisher : Springer
  • Release : 2014-10-08
  • Pages : 344
  • ISBN : 9783709125212
  • Language : En, Es, Fr & De
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The volume aims at giving a comprehensive and up-to-date view of modern methods of analytical mechanics (general equations, invariant objects, stability and bifurcations) and their applications (rigid body dynamics, celestial mechanics, multibody systems etc.). The course is at an advanced level. It is designed for postgraduate students, research engineers and academics that are familiar with basic concepts of analytical dynamics and stability theory. Although the course deals with mechanical problems, most of the concepts and methods involved are equally applicated to general dynamical systems.

Mathematical Methods for Physics

Mathematical Methods for Physics
45th anniversary edition

by H.W. Wyld,Gary Powell

  • Publisher : CRC Press
  • Release : 2020-11-26
  • Pages : 452
  • ISBN : 1000261026
  • Language : En, Es, Fr & De
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From classical mechanics and classical electrodynamics to modern quantum mechanics many physical phenomena are formulated in terms of similar partial differential equations while boundary conditions determine the specifics of the problem. This 45th anniversary edition of the advanced book classic Mathematical Methods for Physics demonstrates how many physics problems resolve into similar inhomogeneous partial differential equations and the mathematical techniques for solving them. The text has three parts: Part I establishes solving the homogenous Laplace and Helmholtz equations in the three main coordinate systems, rectilinear, cylindrical, and spherical and develops the solution space for series solutions to the Sturm-Liouville equation, indicial relations, and the expansion of orthogonal functions including spherical harmonics and Fourier series, Bessel, and Spherical Bessel functions. Many examples with figures are provided including electrostatics, wave guides and resonant cavities, vibrations of membranes, heat flow, potential flow in fluids, and plane and spherical waves. In Part II the inhomogeneous equations are addressed where source terms are included for Poisson's equation, the wave equation, and the diffusion equation. Coverage includes many examples from averaging approaches for electrostatics and magnetostatics, from Green function solutions for time independent and time dependent problems, and from integral equation methods. In Part III complex variable techniques are presented for solving integral equations involving Cauchy Residue theory, contour methods, analytic continuation, and transforming the contour; for addressing dispersion relations; for revisiting special functions in the complex plane; and for transforms in the complex plane including Green’s functions and Laplace transforms. Key Features: · Mathematical Methods for Physics creates a strong, solid anchor of learning and is useful for reference. · Lecture note style suitable for advanced undergraduate and graduate students to learn many techniques for solving partial differential equations with boundary conditions · Many examples across various subjects of physics in classical mechanics, classical electrodynamics, and quantum mechanics · Updated typesetting and layout for improved clarity This book, in lecture note style with updated layout and typesetting, is suitable for advanced undergraduate, graduate students, and as a reference for researchers. It has been edited and carefully updated by Gary Powell.

Mathematical Methods In Classical And Quantum Physics

Mathematical Methods In Classical And Quantum Physics
A Book

by Tulsi Dass,S.K. Sharma

  • Publisher : Universities Press
  • Release : 1998
  • Pages : 703
  • ISBN : 9788173710896
  • Language : En, Es, Fr & De
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This book is intended to provide an adequate background for various theortical physics courses, especially those in classical mechanics, electrodynamics, quatum mechanics and statistical physics. Each topic is dealt with in a generally self-contained manner and the text is interspersed with a number of solved examples ad a large number of exercise problems.

Analytical Mechanics for Relativity and Quantum Mechanics

Analytical Mechanics for Relativity and Quantum Mechanics
A Book

by Oliver Johns

  • Publisher : Oxford University Press
  • Release : 2005-07-07
  • Pages : 597
  • ISBN : 9780198567264
  • Language : En, Es, Fr & De
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This book provides an innovative and mathematically sound treatment of the foundations of analytical mechanics and the relation of classical mechanics to relativity and quantum theory. It is intended for use at the introductory graduate level. A distinguishing feature of the book is its integration of special relativity into teaching of classical mechanics. After a thorough review of the traditional theory, Part II of the book introduces extended Lagrangian and Hamiltonian methods that treat time as a transformable coordinate rather than the fixed parameter of Newtonian physics. Advanced topics such as covariant Langrangians and Hamiltonians, canonical transformations, and Hamilton-Jacobi methods are simplified by the use of this extended theory. And the definition of canonical transformation no longer excludes the Lorenz transformation of special relativity. This is also a book for those who study analytical mechanics to prepare for a critical exploration of quantum mechanics. Comparisons to quantum mechanics appear throughout the text. The extended Hamiltonian theory with time as a coordinate is compared to Dirac's formalism of primary phase space constraints. The chapter on relativisitic mechanics shows how to use covariant Hamiltonian theory to write the Klein-Gordon and Dirac equations. The chapter on Hamilton-Jacobi theory includes a discussion of the closely related Bohm hidden variable model of quantum mechanics. Classical mechanics itself is presented with an emphasis on methods, such as linear vector operators and dyadics, that will familiarize the student with similar techniques in quantum theory. Several of the current fundamental problems in theoretical physics - the development of quantum information technology, and the problem of quantizing the gravitational field, to name two - require a rethinking of the quantum-classical connection. Graduate students preparing for research careers will find a graduate mechanics course based on this book to be an essential bridge between their undergraduate training and advanced study in analytical mechanics, relativity, and quantum mechanics.

Analytical Mechanics for Relativity and Quantum Mechanics

Analytical Mechanics for Relativity and Quantum Mechanics
A Book

by Oliver Johns

  • Publisher : OUP Oxford
  • Release : 2011-05-19
  • Pages : 656
  • ISBN : 0191001627
  • Language : En, Es, Fr & De
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An innovative and mathematically sound treatment of the foundations of analytical mechanics and the relation of classical mechanics to relativity and quantum theory. It presents classical mechanics in a way designed to assist the student's transition to quantum theory.

Classical Mechanics

Classical Mechanics
A Computational Approach with Examples Using Mathematica and Python

by Christopher W. Kulp,Vasilis Pagonis

  • Publisher : CRC Press
  • Release : 2020-11-17
  • Pages : 440
  • ISBN : 135102437X
  • Language : En, Es, Fr & De
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Classical Mechanics: A Computational Approach with Examples using Python and Mathematica provides a unique, contemporary introduction to classical mechanics, with a focus on computational methods. In addition to providing clear and thorough coverage of key topics, this textbook includes integrated instructions and treatments of computation. Full of pedagogy, it contains both analytical and computational example problems within the body of each chapter. The example problems teach readers both analytical methods and how to use computer algebra systems and computer programming to solve problems in classical mechanics. End-of-chapter problems allow students to hone their skills in problem solving with and without the use of a computer. The methods presented in this book can then be used by students when solving problems in other fields both within and outside of physics. It is an ideal textbook for undergraduate students in physics, mathematics, and engineering studying classical mechanics. Features: Gives readers the "big picture" of classical mechanics and the importance of computation in the solution of problems in physics Numerous example problems using both analytical and computational methods, as well as explanations as to how and why specific techniques were used Online resources containing specific example codes to help students learn computational methods and write their own algorithms

A Brief Introduction to Classical, Statistical, and Quantum Mechanics

A Brief Introduction to Classical, Statistical, and Quantum Mechanics
A Book

by Oliver Bühler

  • Publisher : American Mathematical Soc.
  • Release : 2006-10-12
  • Pages : 153
  • ISBN : 0821842323
  • Language : En, Es, Fr & De
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This book provides a rapid overview of the basic methods and concepts in mechanics for beginning Ph.D. students and advanced undergraduates in applied mathematics or related fields. It is based on a graduate course given in 2006-07 at the Courant Institute of Mathematical Sciences. Among other topics, the book introduces Newton's law, action principles, Hamilton-Jacobi theory, geometric wave theory, analytical and numerical statistical mechanics, discrete and continuous quantum mechanics, and quantum path-integral methods. The focus is on fundamental mathematical methods that provide connections between seemingly unrelated subjects. An example is Hamilton-Jacobi theory, which appears in the calculus of variations, in Fermat's principle of classical mechanics, and in the geometric theory of dispersive wavetrains. The material is developed in a sequence of simple examples and the book can be used in a one-semester class on classical, statistical, and quantum mechanics. Some familiarity with differential equations is required but otherwise the book is self-contained. In particular, no previous knowledge of physics is assumed. Titles in this series are co-published with the Courant Institute of Mathematical Sciences at New York University.