Canonical perturbation theories. Degenerate systems and resonance
The book consists of 10 chapters and four appendices. The chapters are: 1. The Hamilton-Jacobi theory. 2. Angle-action variables. Separable systems. 3. Classical perturbation theories. 4. Resonance. 5. Lie mappings. 6. Lie series perturbation theories. 7. Non-singular canonical variables. 8. Lie series theories for resonant systems. 9. Single resonance near singularity. 10. Nonlinear oscillators. The appendices are: A. Bohlin theory. B. The simple pendulum. C. Andoyer Hamiltonian with \(k=1\). D. Andoyer Hamiltonian with \(k \geq 2\). Two first chapters are devoted to Hamilton-Jacobi theory. Chapters 3 and 4 present perturbation theories where canonical transformations are obtained by means of Jacobi generating function. Chapter 5 introduces Lie mappings, and chapter 6 reconsiders perturbed Hamiltonian systems with canonical transformations in the form of Lie series. Chapter 6 discusses also Hori theory with unspecified canonical variables, and chapter 7 demonstrates how Hori theory with unspecified canonical variables allows the construction of formal solutions using non-singular Poincaré solutions, thus allowing the study of perturbed systems near singularities. In chapters 8 and 9, the understanding of the role played by Hori kernel is the key to dealing with resonant systems of different orders. In the final chapter 10, the theories presented in previous chapters are applied to some quasi-harmonic Hamiltonian systems. Additionally, Appendix A presents Bohlin theory and an extension of Delaunay theory. Appendices B and C describe the complete solutions of two integrable Hamiltonians fundamental in resonance studies: the simple pendulum and the first Andoyer Hamiltonian. Finally, appendix D presents the main futures of some higher-order Andoyer Hamiltonians. Undoubtedly, the monograph will be useful for postgraduate students.
scientific article; zbMATH DE number 5081195