| Description |
1 online resource (322 pages) |
| Contents |
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 |
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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 |
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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 |
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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 |
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7.2. The Liouville theorem -- 7.3. The Poincaré 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 |
| Notes |
Print version record |
| Subject |
Mechanics, Analytic -- Mathematics
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| Form |
Electronic book
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| ISBN |
9780128229866 |
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0128229861 |
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