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Group-Theoretic Methods in Mechanics and Applied Mathematics [Kõva köide]

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  • Formaat: Hardback, 240 pages, kõrgus x laius: 254x178 mm, kaal: 600 g
  • Ilmumisaeg: 15-Aug-2002
  • Kirjastus: CRC Press
  • ISBN-10: 0415298636
  • ISBN-13: 9780415298636
Teised raamatud teemal:
Group-Theoretic Methods in Mechanics and Applied MathematicsSuurem pilt
  • Formaat: Hardback, 240 pages, kõrgus x laius: 254x178 mm, kaal: 600 g
  • Ilmumisaeg: 15-Aug-2002
  • Kirjastus: CRC Press
  • ISBN-10: 0415298636
  • ISBN-13: 9780415298636
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9780415298636.html
Märksõnad:
A textbook for graduate and undergraduate students of applied mathematics, mechanics, or physics who have completed the standard introductory courses in advanced calculus and differential equations. It shifts the focus of group theory from the traditional areas of quantum mechanics, crystallography, and nuclear physics to mechanics and applied mathematics. Annotation (c) Book News, Inc., Portland, OR (booknews.com)

Group analysis of differential equations has applications to various problems in nonlinear mechanics and physics. For the first time, this book gives the systematic group analysis of main postulates of classical and relativistic mechanics. The consistent presentation of Lie group theory is illustrated by plentiful examples. Symmetries and conservation laws of differential equations are studied. Specific equations and problems of mechanics and physics are considered, and exact solutions are given for the following equations: dynamics of rigid body, heat transfer, wave, hydrodynamics, Thomas-Fermi and more. The author pays particular attention to the application of group analysis to developing asymptotic methods of applied mathematics in problems with small parameter. The methods are used to solve basic equations (Van Der Pol's equation, Duffing equation, etc.) encountered in the theory of nonlinear oscillations. This book is intended for a wide range of scientists, engineers and students in the fields of applied mathematics, mechanics and physics.
Foreword vii
Authors x
Basic Notions of Lie Group Theory
1(46)
Notion of Group
1(2)
Lie Group. Examples
3(2)
Group Generator. Lie Algebra
5(6)
One-Parameter Groups. Uniqueness Theorem
11(2)
Liouville Equation. Invariants. Eigenfunctions
13(4)
Linear Partial Differential Equations
17(1)
Change of Variables. Canonical Coordinates of a Group
18(2)
Hausdorff's Formula. Symmetry Groups
20(9)
Principle of Superposition of Solutions and Separation of Motions in Nonlinear Mechanics
29(2)
Prolongation of Groups. Differential and Integral Invariants
31(5)
Equations Admitting a Given Group
36(6)
Symmetries of Partial Differential Equations
42(5)
Group Analysis of Foundations of Classical and Relativistic Mechanics
47(18)
Axiomatization Problem of Mechanics
47(1)
Postulates of Classical Mechanics
48(2)
Projective Symmetries of Newton's First Law
50(1)
Newton's Second Law. Galilean Symmetries
51(3)
Postulates of Relativistic Mechanics
54(1)
Group of Symmetries of Maxwell's Equations
54(2)
Twice Prolonged Lorentz Group
56(4)
Differential and Integral Invariants of the Lorentz Group
60(2)
Relativistic Equations of Motion of a Particle
62(1)
Noninertial Reference Frames
63(2)
Application of Group Methods to Problems of Mechanics
65(60)
Perturbation Theory for Configuration Manifolds of Resonant Systems
65(8)
Statement of the Problem
65(1)
The Case of Double Natural Frequency
66(1)
The Manifold of Degenerate Forms. Local Evolution Basis
67(2)
Algebra of Local Evolutions
69(1)
Classification of Perturbations
70(1)
The Problem of Stabilization of the Oscillation Shape
71(2)
Poincare's Equation on Lie Algebras
73(3)
Kinematics of a Rigid Body
76(28)
Ways of Specifying the Orientation of a Rigid Body
76(14)
Addition of Rotations
90(6)
Topology of the Manifold of Rotations of a Rigid Body (Topology of the SO(3) group)
96(1)
Angular Velocity of a Rigid Body
97(7)
Problems of Mechanics Admitting Similarity Groups
104(8)
Suslov Problem
104(3)
The Problem of the Follower Trajectory
107(2)
Rolling of a Homogeneous Ball Over a Rough Plane
109(3)
Problems With Determinable Linear Groups of Symmetries
112(13)
Falling of a Heavy Homogeneous Thread
112(3)
Motion of a Point Particle Under the Action of a Follower Force
115(5)
The Problem of an Optimal Shape of a Body in an Air Flow
120(5)
Finite-Dimensional Hamiltonian Systems
125(26)
Legendre Transformation
125(1)
Hamiltonian Systems. Poisson Bracket
126(2)
Nonautonomous Hamiltonian Systems
128(1)
Integrals of Hamiltonian Groups. Noether's Theorem
129(3)
Conservation Laws and Symmetries
132(1)
Integral Invariants
133(6)
Poincare--Cartan Invariants
134(4)
Liouville's Theorem of the Phase Volume
138(1)
Canonical Transformations
139(6)
Hamilton-Jacobi Equation
145(2)
Liouville's Theorem of Integrable Systems
147(1)
The Angle-Action Variables
148(3)
Asymptotic Methods of Applied Mathematics
151(68)
Introduction
151(2)
Normal Coordinates of Conservative Systems
153(5)
Single-Frequency Method of Averaging Based on Hausdorff's Formula
158(2)
Poincare Normal Form
160(10)
The Averaging Principle
170(19)
Averaging of Single-Frequency Systems
175(8)
Multifrequency Systems. Resonance
183(6)
Asymptotic Integration of Hamiltonian Systems
189(15)
Birkhoff Normal Form
189(11)
Averaging of Hamiltonian Systems in Terms of Lie Series
200(1)
Artificial Hamiltonization
201(3)
Method of Tangent Approximations
204(4)
Classical Examples of Oscillation Theory
208(11)
Van Der Pol's Equation
208(1)
Mathieu's Equation
209(3)
Forced Oscillations of Duffing's Oscillator
212(3)
Forced Oscillations of Van Der Pol's Oscillator
215(4)
Brief Historical Sketch 219(6)
Index 225
Klimov, D.M. | Zhuravlev, V. Ph.