Perturbation Methods in Non-Linear Systems - Georgio Eugenio Oscare Giacaglia

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Sommaire:
I. Canonical Transformation Theory and Generalizations.- 1. Introduction.- 2. Canonical Transformations.- 3. Hamilton-Jacobi Equation. Generalizations.- 4. Lie Series and Lie Transforms.- 5. Lie Transform Depending on a Parameter.- 6. Equivalence Relations.- 7. General Transformations Induced by Lie Series.- Notes.- References.- II. Perturbation Methods for Hamiltonian Systems. Generalizations.- 1. Introduction.- 2. Convergence of a Classical Method of Iteration.- 3. Secular Terms. Lindstedt's Device.- 4. Poincar?'s Method (Lindstedt's Method).- 5. Fast and Slow Variables.- 6. Generalization of the Averaging Procedure, Birkoff's Normalization and Adelphic Integrals.- 7. The Solution of Poincar?'s Problem in Poisson's Parentheses. Elimination of Secular Terms from Adelphic Integrals.- 8. Perturbation Techniques Based on Lie Transforms.- 9. Perturbation Methods of Non-Hamiltonian Systems Based on Lie Transforms.- Notes.- References.- III. Perturbations of Integrable Systemsl.- 1. Motion of an Integrable System.- 2. Perturbations of an Integral System.- 3. Degenerate Systems.- 4. Perturbed Linear Oscillations.- 5. Linear Periodic Perturbations.- Notes.- References.- IV. Perturbations of Area Preserving Mappings.- 1. Preliminary Considerations.- 2. Regions of Motion. Perturbation of a Truncated Birkoff's Normal Form.- 3.Moser's Theorem.- 4. System with n Degree of Freedom.- 5. Degenerate Systems.- Notes.- References.- V. Resonance.- 1. Introduction.- 2. Motion in the Neighborhood of an Equilibrium Point.- 3. Solution by Formal Series28l.- 4. Equivalence with the Problem of Perturbation of a Linear System.- 5. Nonlinear Resonance.- 6. Asymptotic Expansion to Any Order.- 7. Extended Theory and the Ideal Resonance Problem.- 8. Several Degrees of Freedom.- 9.Coupling of Two Harmonic Oscillators.- Notes.- References.- Appendix. Remarks, Some Open Questions and Research Topics.- References.