2024/2025
Введение в КАМ-теорию
Статус:
Дисциплина общефакультетского пула
Кто читает:
Факультет математики
Где читается:
Факультет математики
Когда читается:
3, 4 модуль
Охват аудитории:
для всех кампусов НИУ ВШЭ
Преподаватели:
Глуцюк Алексей Антонович
Язык:
английский
Кредиты:
3
Course Syllabus
Abstract
Hamilton equations are one of the most fundamental class of differential equations describing law of movements of mechanical systems and many physical laws. Their flows preserve symplectic structure and hence, the corresponding volume form. The simplest class of Hamiltonian systems are integrable Hamiltonian systems on compact symplectic manifolds. In this case Arnold -- Liouville Theorem states that the manifold is fibered by invariant half-dimensional tori along which the movement is quasi-periodic: its trajectories follow integral curves of linear flow on torus. The KAM theory created by A.N.Kolmogorov, V.I.Arnold and J.Moser in late 1950-ths - early 1960-ths deals with perturbations of the so-called non-degenerate integrable Hamiltonian systems: namely, those of them, where the frequency vector of quasi-periodic movement along an invariant torus has non-degenerate derivative in the transversal parameter. The main KAM theorem states that for every small perturbation most of invariant tori survive: the smaller is the perturbation parameter, the more percentage (in the sense of Lebesgue measure) of survived tori. The KAM theory is used in many domains of mathematics, mechanics and physics. First of all, in symplectic dynamics and in celestial mechanics. And also in the theory of billiards. The goal of the cours is to present the KAM theory with a proof of the KAM theorem and its different versions, and to discuss its applications and related questions. Including the case of perturbations of integrable twist symplectomorphisms of planar cylinder (persistance of most of invariant circles) and billiard theory (Lazutkin's Theorem on existence of Cantor family (of positive measure) of caustics: curves whose tangent lines are reflected again to their tangent lines). We will also discuss global dynamics of the perturbed system; first of all, for perturbations of integrable twist symplectomorphism of cylinder.
Course Contents
- Hamiltonian systems. Examples from mechanics. Invariance of symplectic form and the corresponding volume.
- Symplectic manifolds and Hamiltonian vector fields on them. Integrable Hamiltonian systems. Arnold – Liouville Theorem. Classical examples.
- Symplectomorphisms. Integrable twist symplectomorphisms of planar cylinder. KAM theorem for their perturbations and dynamics of the perturbed symplectomorphism.
- KAM Theorem for general non-degenerate integrable Hamiltonian system
- Symplectic structure on the space of oriented lines in Euclidean space. Billiard reflections of oriented lines as symplectomorphisms. Generalization to geodesics and billiards on arbitrary Riemannian manifold: Melrose reduction.
- Convex planar billiards. Caustics. Integrability of elliptic billiard
- Lazutkin KAM type Theorem on convex planar billiards: existence of Cantor family (of positive measure) of disjoint closed caustics
- A survey of Birkhoff Conjecture on integrable billiards
Interim Assessment
- 2024/2025 4th module0.6𝑥 (grade for problem solutions) +0.6𝑥 (exam grade). If the number thus obtained is greater than 10, then the final grade is 10.
