Mathematical Methods of Analytical Mechanics 1st Edition by Henri Gouin 0128229861 9780128229866

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ISBN-10 :  0128229861 

ISBN-13 :  9780128229866

Author : Henri Gouin 

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.

Mathematical Methods of Analytical Mechanics 1st Table of contents

Part 1: Introduction to the Calculus of Variations
1: Elementary Methods to the Calculus of Variations
Abstracts
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
Abstracts
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
Abstracts
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
Abstracts
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
Abstracts
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
Abstracts
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
Abstracts
7.1: Flow of a dynamical system
7.2: The Liouville theorem
7.3: The Poincaré recurrence theorem
8: Oscillations and Small Motions of Mechanical Systems
Abstracts
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
Abstracts
9.1: Position of the problem
9.2: Flow of a periodic differential equation
9.3: Study of the planar case
9.4: Strong stability in periodic Hamiltonian systems
9.5: Study of the Mathieu equation. Parametric resonance
9.6: A completely integrable case of the Hill equation
Part 4: Problems and Exercises
Introduction
10: Problems and Exercises

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