Quasi-integrable mechanical systems. (English) Zbl 0662.70022
Critical phenomena, random systems, gauge theories, Proc. Summer Sch. Theor. Phys., Sess. 43, Les Houches/France 1984, Pt. 2, 539-624 (1986).
[For the entire collection see Zbl 0651.00019.]
An exposition of some topics of modern integrability theory of mechanical systems with finite-dimensional phase space. The following problems are under discussion: 1. Basic definitions and canonical integrability with illustrative examples (the system of free rotators, the two-body problem, a solid with a fixed point, the geodesic motion on an ellipsoid). 2. Canonical integrability and the Arnold-Liouville theorem. Interrelations between integrability and quasi-periodicity of all motions. 3. Classical perturbation theory. 4. Birkhoff theorem on harmonic oscillators. The problem of convergence of Birkhoff series. 5. Some applications. The calculation of the precession of Mercury under the influence of Jupiter. The Poincaré triviality theorem referring to nonexistence of analytically dependent constants of motion other then the energy. 6. Phase-space diffusion. 7. Resonances and chaos. The forced pendulum motion as an illustration of how chaotic motions arise. 8. Existence of nonresonant tori and quasi-periodic motions. The proof of Kolmogorov- Arnold-Moser theorem. 9. Concluding remarks and some author’s results referring to KAM theory.
An exposition of some topics of modern integrability theory of mechanical systems with finite-dimensional phase space. The following problems are under discussion: 1. Basic definitions and canonical integrability with illustrative examples (the system of free rotators, the two-body problem, a solid with a fixed point, the geodesic motion on an ellipsoid). 2. Canonical integrability and the Arnold-Liouville theorem. Interrelations between integrability and quasi-periodicity of all motions. 3. Classical perturbation theory. 4. Birkhoff theorem on harmonic oscillators. The problem of convergence of Birkhoff series. 5. Some applications. The calculation of the precession of Mercury under the influence of Jupiter. The Poincaré triviality theorem referring to nonexistence of analytically dependent constants of motion other then the energy. 6. Phase-space diffusion. 7. Resonances and chaos. The forced pendulum motion as an illustration of how chaotic motions arise. 8. Existence of nonresonant tori and quasi-periodic motions. The proof of Kolmogorov- Arnold-Moser theorem. 9. Concluding remarks and some author’s results referring to KAM theory.
Reviewer: I.Dorfman
MSC:
| 70K30 | Nonlinear resonances for nonlinear problems in mechanics |
| 37J35 | Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests |
| 37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |
| 37C55 | Periodic and quasi-periodic flows and diffeomorphisms |
| 37J40 | Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion |
