Group-Theoretic Methods in Mechanics and Applied Mathematics.

Cover -- Half Title -- Series Page -- Title Page -- Copyright Page -- Contents -- Foreword -- Authors -- Chapter 1. Basic Notions of Lie Group Theory -- 1.1. Notion of Group -- 1.2. Lie Group. Examples -- 1.3. Group Generator. Lie Algebra -- 1.4. One-Parameter Groups. Uniqueness Theorem -- 1.5. Liouville Equation. Invariants. Eigenfunctions -- 1.6. Linear Partial Differential Equations -- 1. 7. Change of Variables. Canonical Coordinates of a Group -- 1.8. Hausdorff's Formula. Symmetry Groups -- 1.9. Principle of Superposition of Solutions and Separation of Motions in Nonlinear Mechanics -- 1.1 0. Prolongation of Groups. Differential and Integral Invariants -- 1.11. Equations Admitting a Given Group -- 1.12. Symmetries of Partial Differential Equations -- Chapter 2. Group Analysis of Foundations of Classical and Relativistic Mechanics -- 2.1. Axiomatization Problem of Mechanics -- 2.2. Postulates of Classical Mechanics -- 2.3. Projective Symmetries of Newton's First Law -- 2.4. Newton's Second Law. Galilean Symmetries -- 2.5. Postulates of Relativistic Mechanics -- 2.6. Group of Symmetries of Maxwell's Equations -- 2.7. Twice Prolonged Lorentz Group -- 2.8. Differential and Integral Invariants of the Lorentz Group -- 2.9. Relativistic Equations of Motion of a Particle -- 2.1 0. Noninertial Reference Frames -- Chapter 3. Application of Group Methods to Problems of Mechanics -- 3.1. Perturbation Theory for Configuration Manifolds of Resonant Systems -- 3.1.1. Statement of the Problem -- 3.1.2. The Case of Double Natural Frequency -- 3.1.3. The Manifold of Degenerate Forms. Local Evolution Basis -- 3.1.4. Algebra of Local Evolutions -- 3.1.5. Classification of Perturbations -- 3.1.6. The Problem of Stabilization of the Oscillation Shape -- 3.2. Poincare's Equation on Lie Algebras -- 3.3. Kinematics of a Rigid Body.

3.3.1. Ways of Specifying the Orientation of a Rigid Body -- 3.3.2. Addition of Rotations -- 3.3.3. Topology of the Manifold of Rotations of a Rigid Body (Topology of the S0(3) group -- 3.3.4. Angular Velocity of a Rigid Body -- 3.4. Problems of Mechanics Admitting Similarity Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . I -- 3.4.1. Suslov Problem -- 3.4.2. The Problem of the Follower Trajectory -- 3.4.3. Rolling of a Homogeneous Ball Over a Rough Plane -- 3.5. Problems With Determinable Linear Groups of Symmetries -- 3.5.1. Falling of a Heavy Homogeneous Thread -- 3.5.2. Motion of a Point Particle Under the Action of a Follower Force -- 3.5.3. The Problem of an Optimal Shape of a Body in an Air Flow -- Chapter 4. Finite-Dimensional Hamiltonian Systems -- 4.1. Legendre Transformation -- 4.2. Hamiltonian Systems. Poisson Bracket -- 4.3. Nonautonomous Hamiltonian Systems -- 4.4. Integrals of Hamiltonian Groups. Noether's Theorem -- 4.5. Conservation Laws and Symmetries -- 4.6. Integral Invariants -- 4.6.1. Poincare--Cartan Invariants -- 4.6.2. Liouville's Theorem of the Phase Volume -- 4.7. Canonical Transformations -- 4.8. Hamilton-Jacobi Equation -- 4.1 0. The Angle-Action Variables -- Chapter 5. Asymptotic Methodsof Applied Mathematics -- 5.1. Introduction -- 5.2. Normal Coordinates of Conservative Systems -- 5.3. Single-Frequency Method of Averaging Based onHausdorff's Formula -- 5.4. Poincare Normal Form -- 5.5. The Averaging Principle -- 5.5.1. Averaging of Single-Frequency Systems -- 5.5.2. Multifrequency Systems. Resonance -- 5.6. Asymptotic Integration of Hamiltonian Systems -- 5.6.1. Birkhoff Normal Form -- 5.6.2. Averaging of Hamiltonian Systems in Terms of Lie Series -- 5.6.3. Artificial Hamiltonization -- 5. 7. Method of Tangent Approximations -- 5.8. Classical Examples of Oscillation Theory -- 5.8.1. Van Der Pol's Equation.

5.8.2. Mathieu's Equation -- 5.8.3. Forced Oscillations of Cuffing's Oscillator -- 5.8.4. Forced Oscillations of Van Der Pol's Oscillator -- Brief Historical Sketch -- Index.

Group analysis of differential equations has applications to various problems in nonlinear mechanics and physics. Group-Theoretic Methods in Mechanics and Applied Mathematics systematizes the group analysis of the main postulates of classical and relativistic mechanics. 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 for problems with small parameters. This book is designed for a broad audience of scientists, engineers, and students in the fields of applied mathematics, mechanics and physics.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.