Group-theoretic methods in mechanics and applied mathematics

D.M. Klimov, V.Ph. Zhuravlev

240 pages, parution le 14/10/2002

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Résumé

This volume :

Presents the basics of group analysis of differential equations
Performs group-theoretic analysis of the axioms of classical and relativistic mechanics
Outlines group-theoretic foundations of kinematics of rotations, Poincaré's equations, Hamiltonian mechanics, theory of resonant systems and others
Gives solutions of a number of new mechanical problems
Describes group-theoretic methods for the construction of asymptotic expansions in applied mathematics
May be used by lecturers at universities and colleges as a basis for courses on mechanics and applied mathematics

The volume is intented for researchers, university teachers, engineers and students in mechanics, applied mathematics, physics and engineering sciences.ContentsI- Basic Notions of Lie Group Theory

Notion of Group
Lie Group. Examples
Group Generator. Lie Algebra
One-Parameter Groups. Uniqueness Theorem
Liouville Equation. Invariants. Eigenfunctions
Linear Partial Differential Equations
Change of Variables. Canonical Coordinates of a Group
Hausdorff's Formula. Symmetry Groups
Principle of Superposition of Solutions and Separation of Motions in Nonlinear Mechanics
Prolongation of Groups. Differential and Integral Invariants
Equations Admitting a Given Group
Symmetries of Partial Differential Equations

II- Group Analysis of Foundations of Classical and Relativistic Mechanics

Axiomatization Problem of Mechanics
Postulates of Classical Mechanics
Projective Symmetries of Newton's First Law
Newton's Second Law. Galilean Symmetries
Postulates of Relativistic Mechanics
Group of Symmetries of Maxwell's Equations
Twice Prolonged Lorentz Group
Differential and Integral Invariants of the Lorentz Group
Relativistic Equations of Motion of a Particle
Noninertial Reference Frames

III- Application of Group Methods to Problems of Mechanics

Perturbation Theory for Configuration Manifolds of Resonant Systems
Poincare's Equation on Lie Algebras
Kinematics of a Rigid Body
Problems of Mechanics Admitting Similarity Groups
Problems With Determinable Linear Groups of Symmetries

IV- Finite-Dimensional Hamiltonian Systems

Legendre Transformation
Hamiltonian Systems. Poisson Bracket
Nonautonomous Hamiltonian Systems
Integrals of Hamiltonian Groups. Noether's Theorem
Conservation Laws and Symmetries
Integral Invariants
Canonical Transformations
Hamilton-Jacobi Equation
Liouville's Theorem of Integrable Systems
The Angle-Action Variables

V- Asymptotic Methods of Applied Mathematics

Introduction
Normal Coordinates of Conservative Systems
Single-Frequency Method of Averaging Based on Hausdorff's Formula
Poincare Normal Form
The Averaging Principle
Asymptotic Integration of Hamiltonian Systems
Method of Tangent Approximations
Classical Examples of Oscillation Theory

Caractéristiques techniques

	PAPIER
Éditeur(s)	Taylor and Francis Books
Auteur(s)	D.M. Klimov, V.Ph. Zhuravlev
Parution	14/10/2002
Nb. de pages	240
Format	17,7 x 25,2
Couverture	Relié
Poids	576g
Intérieur	Noir et Blanc
EAN13	9780415298636
ISBN13	978-0-415-29863-6