13. Introduction to KAM theory with a view to celestial mechanics
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Jacques Féjoz
Abstract
The theory of Kolmogorov, Arnold, and Moser (KAM) consists of a set of results regarding the persistence of quasiperiodic solutions, primarily in Hamiltonian systems.We bring forward a “twisted conjugacy” normal form, due to Herman, which contains all the (not so) hard analysis. We focus on the real analytic setting. A variety of KAM results follow, includingmost classical statements as well asmore general ones. This strategy makes it simple to deal with various kinds of degeneracies and symmetries. As an example of application, we prove the existence of quasiperiodic motions in the spatial lunar three-body problem
Abstract
The theory of Kolmogorov, Arnold, and Moser (KAM) consists of a set of results regarding the persistence of quasiperiodic solutions, primarily in Hamiltonian systems.We bring forward a “twisted conjugacy” normal form, due to Herman, which contains all the (not so) hard analysis. We focus on the real analytic setting. A variety of KAM results follow, includingmost classical statements as well asmore general ones. This strategy makes it simple to deal with various kinds of degeneracies and symmetries. As an example of application, we prove the existence of quasiperiodic motions in the spatial lunar three-body problem
Kapitel in diesem Buch
- Frontmatter I
- Contents V
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Part I
- 1. Second-order decomposition model for image processing: numerical experimentation 5
- 2. Optimizing spatial and tonal data for PDE-based inpainting 35
- 3. Image registration using phase–amplitude separation 84
- 4. Rotation invariance in exemplar-based image inpainting 108
- 5. Convective regularization for optical flow 184
- 6. A variational method for quantitative photoacoustic tomography with piecewise constant coefficients 202
- 7. On optical flow models for variational motion estimation 225
- 8. Bilevel approaches for learning of variational imaging models 252
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Part II
- 9. Non-degenerate forms of the generalized Euler–Lagrange condition for state-constrained optimal control problems 295
- 10 The Purcell three-link swimmer: some geometric and numerical aspects related to periodic optimal controls 314
- 11. Controllability of Keplerian motion with low-thrust control systems 344
- 12. Higher variational equation techniques for the integrability of homogeneous potentials 365
- 13. Introduction to KAM theory with a view to celestial mechanics 387
- 14. Invariants of contact sub-pseudo-Riemannian structures and Einstein–Weyl geometry 434
- 15. Time-optimal control for a perturbed Brockett integrator 454
- 16. Twist maps and Arnold diffusion for diffeomorphisms 473
- 17. A Hamiltonian approach to sufficiency in optimal control with minimal regularity conditions: Part I 496
- Index 517
Kapitel in diesem Buch
- Frontmatter I
- Contents V
-
Part I
- 1. Second-order decomposition model for image processing: numerical experimentation 5
- 2. Optimizing spatial and tonal data for PDE-based inpainting 35
- 3. Image registration using phase–amplitude separation 84
- 4. Rotation invariance in exemplar-based image inpainting 108
- 5. Convective regularization for optical flow 184
- 6. A variational method for quantitative photoacoustic tomography with piecewise constant coefficients 202
- 7. On optical flow models for variational motion estimation 225
- 8. Bilevel approaches for learning of variational imaging models 252
-
Part II
- 9. Non-degenerate forms of the generalized Euler–Lagrange condition for state-constrained optimal control problems 295
- 10 The Purcell three-link swimmer: some geometric and numerical aspects related to periodic optimal controls 314
- 11. Controllability of Keplerian motion with low-thrust control systems 344
- 12. Higher variational equation techniques for the integrability of homogeneous potentials 365
- 13. Introduction to KAM theory with a view to celestial mechanics 387
- 14. Invariants of contact sub-pseudo-Riemannian structures and Einstein–Weyl geometry 434
- 15. Time-optimal control for a perturbed Brockett integrator 454
- 16. Twist maps and Arnold diffusion for diffeomorphisms 473
- 17. A Hamiltonian approach to sufficiency in optimal control with minimal regularity conditions: Part I 496
- Index 517
