Notes on Hamiltonian dynamical systems
著者
書誌事項
Notes on Hamiltonian dynamical systems
（London Mathematical Society student texts, 102）
Cambridge University Press, 2022
- : hardback
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注記
Includes bibliographical references (p. [429]-443) and index
内容説明・目次
内容説明
Starting with the basics of Hamiltonian dynamics and canonical transformations, this text follows the historical development of the theory culminating in recent results: the Kolmogorov-Arnold-Moser theorem, Nekhoroshev's theorem and superexponential stability. Its analytic approach allows students to learn about perturbation methods leading to advanced results. Key topics covered include Liouville's theorem, the proof of Poincare 's non-integrability theorem and the nonlinear dynamics in the neighbourhood of equilibria. The theorem of Kolmogorov on persistence of invariant tori and the theory of exponential stability of Nekhoroshev are proved via constructive algorithms based on the Lie series method. A final chapter is devoted to the discovery of chaos by Poincare and its relations with integrability, also including recent results on superexponential stability. Written in an accessible, self-contained way with few prerequisites, this book can serve as an introductory text for senior undergraduate and graduate students.
目次
- 1. Hamiltonian formalism
- 2. Canonical transformations
- 3. Integrable systems
- 4. First integrals
- 5. Nonlinear oscillations
- 6. The method of Lie series and of Lie transform
- 7. The normal form of Poincare and Birkhoff
- 8. Persistence of invariant tori
- 9. Long time stability
- 10. Stability and chaos
- A. The geometry of resonances
- B. A quick introduction to symplectic geometry
- References
- Index.
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