Mathematical Methods of Analytical Mechanics

Detalles Bibliográficos
Clasificación:Libro Electrónico
Autor principal: Gouin, Henri
Formato: Electrónico eBook
Idioma:Inglés
Publicado: San Diego : ISTE Press Ltd. : Elsevier, 2020.
Temas:
Mechanics, Analytic > Mathematics.
M�ecanique analytique > Math�ematiques.
Acceso en línea:Texto completo
Tabla de Contenidos:
  • Front Cover
  • Introduction to Mathematical Methods of Analytical Mechanics
  • Copyright Page
  • Contents
  • Preface
  • Mathematicians, Physicists and Astronomers Cited in this Book
  • Important Notations
  • PART 1: Introduction to the Calculus of Variations
  • 1. Elementary Methods to the Calculus of Variations
  • 1.1. First free extremum problems
  • 1.2. First constrained extremum problem
  • Lagrange multipliers
  • 1.3. The fundamental lemma of the calculus of variations
  • 1.4. Extremum of a free functional
  • 1.5. Extremum for a constrained functional
  • 1.6. More general problem of the calculus of variations
  • 2. Variation of Curvilinear Integral
  • 2.1. Geometrization of variational problems
  • 2.2. First form of curvilinear integral
  • 2.3. Second form of curvilinear integrals
  • 2.4. Generalization and variation of derivative
  • 2.5. First application: studying the optical path of light
  • 2.6. Second application: the problem of isoperimeters
  • 3. The Noether Theorem
  • 3.1. Additional results on differential equations
  • 3.2. One-parameter groups and Lie groups
  • 3.3. Invariant integral under a Lie group
  • 3.4. Further examination of Fermat's principle
  • PART 2: Applications to Analytical Mechanics
  • 4. The Methods of Analytical Mechanics
  • 4.1. D'Alembert's principle
  • 4.2. Back to analytical mechanics
  • 4.3. The vibrating strings
  • 4.4. Homogeneous Lagrangian. Expression in space time
  • 4.5. The Hamilton equations
  • 4.6. First integral by using the Noether theorem
  • 4.7. Re-injection of a partial result
  • 4.8. The Maupertuis principle
  • 5. Jacobi's Integration Method
  • 5.1. Canonical transformations
  • 5.2. The Jacobi method
  • 5.3. The material point in various systems of representation
  • 5.4. Case of the Liouville integrability
  • 5.5. A specific change of canonical variables
  • 5.6. Multi-periodic systems. Action variables
  • 6. Spaces of Mechanics
  • Poisson Brackets
  • 6.1. Spaces in analytical mechanics
  • 6.2. Dynamical variables
  • Poisson brackets
  • 6.3. Poisson bracket of two dynamical variables
  • 6.4. Canonical transformations
  • 6.5. Remark on the symplectic scalar product
  • PART 3: Properties of Mechanical Systems
  • 7. Properties of Phase Space
  • 7.1. Flow of a dynamical system
  • 7.2. The Liouville theorem
  • 7.3. The Poincar�e recurrence theorem
  • 8. Oscillations and Small Motions of Mechanical Systems
  • 8.1. Preliminary remarks
  • 8.2. The Weierstrass discussion
  • 8.3. Equilibrium position of an autonomous differential equation
  • 8.4. Stability of equilibrium positions of an autonomous differential equation
  • 8.5. A necessary condition of stability
  • 8.6. Linearization of a differential equation
  • 8.7. Behavior of eigenfrequencies
  • 8.8. Perturbed equation associated with linear differential equation
  • 9. The Stability of Periodic Systems

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