Topological Classification of Integrable Systems ( Advances in Soviet Mathematics )

Publication series ： Advances in Soviet Mathematics

Author： A. T. Fomenko  

Publisher： American Mathematical Society‎

Publication year： 2018

E-ISBN: 9781470445539

P-ISBN(Paperback): 9780821841051

Subject： O1 Mathematics

Keyword： 数学

Language： ENG

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Topological Classification of Integrable Systems

Description

In recent years, researchers have found new topological invariants of integrable Hamiltonian systems of differential equations and have constructed a theory for their topological classification. Each paper in this important collection describes one of the “building blocks” of the theory, and several of the works are devoted to applications to specific physical equations. In particular, this collection covers the new topological invariants of integrable equations, the new topological obstructions to integrability, a new Morse-type theory of Bott integrals, and classification of bifurcations of the Liouville tori in integrable systems. The papers collected here grew out of the research seminar “Contemporary Geometrical Methods” at Moscow University, under the guidance of A. T. Fomenko, V. V. Trofimov, and A. V. Bolsinov. Bringing together contributions by some of the experts in this area, this collection is the first publication to treat this theory in a comprehensive way.

Chapter

Cover

Title page

Contents

Introduction

Connections of the Theory of Topological Classification of Integrable Hamiltonian Systems of Differential Equations with Different Geometrical and Topological Problems

The theory of invariants of multidimensional integrable Hamiltonian systems (with arbitrary many degrees of freedom). Molecular table of all integrable systems with two degrees of freedom

Integrable Hamiltonian systems in analytic dynamics and mathematical physics

Fomenko invariants for the main integrable cases of the rigid body motion equations

Methods of calculation of the Fomenko-Zieschang invariant

Topological invariants for some algebraic analogs of the Toda lattice

Topological classification of integrable Bott geodesic flows on the two-dimensional torus

On the complexity of integrable Hamiltonian systems on three-dimensional isoenergy submanifolds

Symplectic connections and Maslov-Arnold characteristic classes

Topological classification of integrable nondegenerate Hamiltonians on the isoenergy three-dimensional sphere

Description of the structure of Fomenko invariants on the boundary and inside 𝑄-domains, estimates of their number on the lower boundary for the manifolds 𝑆³, ℝℙ³, 𝕊¹×𝕊², and 𝕋³

Theory of rough classification of integrable nondegenerate Hamiltonian differential equations on four-dimensional manifolds. Application to classical mechanics

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