Tutorial: KAM theory and the stability of the Solar System

Описание: This lecture explains what the famous Kolmogorov-Arnold-Moser (KAM) theorem on perturbed integrable Hamiltonian system says, and gives an idea of its proof in the particular case of two-dimensional, area-preserving maps. One of the reasons for developing this theorem (but by far not its only application) is the question of the stability of our Solar System. We will discuss whether the KAM theorem has something to say on that - or not.

Intro: 0:00
Why shouldn't the Solar System be stable? 0:42
Billiards: 5:59
The standard map: 10:42
Invariant curves: 13:28
Small divisors: 26:00
Diophantine numbers: 31:05
Back to invariant curves: 48:08
Newton's method: 51:17
The KAM theorem: 57:50
Stability of elliptic fixed points: 1:02:54
Is the Solar System stable? 1:09:00

Tutorial on stability of periodic orbits:    • Tutorial: Stability of periodic orbits - a...  
Related lecture notes: https://arxiv.org/abs/math.HO/0111178

Literature:
A. N. Kolmogorov, "On the Conservation of Conditionally Periodic Motions under Small Perturbation of the Hamiltonian [О сохранении условнопериодических движений при малом изменении функции Гамильтона]," Dokl. Akad. Nauk SSR 98 (1954).
J. Moser, "On invariant curves of area-preserving mappings of an annulus," Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II 1962 (1962), 1–20.
V. I. Arnold, "Proof of a theorem of A. N. Kolmogorov on the preservation of conditionally periodic motions under a small perturbation of the Hamiltonian [Малые знаменатели и проблема устойчивости движения в классической и небесной механике]," Uspekhi Mat. Nauk 18 (1963) (English transl.: Russ. Math. Surv. 18, 9--36, doi:10.1070/RM1963v018n05ABEH004130 - https://iopscience.iop.org/article/10... ).
J. Laskar, "A numerical experiment on the chaotic behaviour of the Solar System", Nature, vol. 338, no 6212, 989, p. 237–238 (DOI 10.1038/338237a0 - https://dx.doi.org/10.1038/338237a0 )
Jacques Laskar and M. Gastineau, "Existence of collisional trajectories of Mercury, Mars and Venus with the Earth", Nature, vol. 459, 2009, p. 817-819 (DOI 10.1038/nature08096 - https://dx.doi.org/10.1038/nature08096 )

#KAM_theory #solarsystem #stability #chaos