Qualitative geometrical theory of integrable systems. classification of isoenergetic surfaces and bifurcation of liouville tori at the critical energy values

1988 ◽  
pp. 221-245 ◽  
Author(s):  
A. T. Fomenko
Keyword(s):  
Integrable Systems ◽  
Critical Energy ◽  
Geometrical Theory ◽  
Liouville Tori ◽  
Bifurcation Of Liouville Tori

scholarly journals Geminography: the crystallography of twins

2006 ◽  
Vol 221 (1) ◽  
Author(s):  
Hans Grimmer ◽  
Massimo Nespolo
Keyword(s):  
Subgroup Analysis ◽  
Geometric Theory ◽  
Reciprocal Space ◽  
Geometrical Theory ◽  
Special Cases ◽  
Recent Developments

AbstractThe geometric theory of twinning was developed almost a century ago. Despite its age, it still represents the fundamental approach to the analysis and interpretation of twinned crystals, in both the direct and the reciprocal space. In recent years, this theory has been extended not only in its formalism (group-subgroup analysis, chromatic symmetry) but also in its classification of special cases that were not recognized before. The geometrical theory of twinning is thus reviewed here with emphasis on lattice aspects and recent developments. The classification of various types of twins starts with Friedel’s well-known scheme, which distinguishes four cases according to whether the twin index

scholarly journals Formal Pseudodifferential Operators in One and Several Variables, Central Extensions, and Integrable Systems

2015 ◽  
Vol 2015 ◽  
pp. 1-16 ◽  
Author(s):  
Jarnishs Beltran ◽  
Enrique G. Reyes
Keyword(s):  
Lie Algebras ◽  
Nonlinear Equations ◽  
Integrable Systems ◽  
Differential Operators ◽  
Pseudodifferential Operators ◽  
Central Extensions ◽  
Several Variables ◽  
Zero Curvature ◽  
Manin Triples

We review some aspects of the theory of Lie algebras of (twisted and untwisted) formal pseudodifferential operators in one and several variables in a general algebraic context. We focus mainly on the construction and classification of nontrivial central extensions. As applications, we construct hierarchies of centrally extended Lie algebras of formal differential operators in one and several variables, Manin triples and hierarchies of nonlinear equations in Lax and zero curvature form.

scholarly journals On the integrability of Birkhoff billiards

2018 ◽  
Vol 376 (2131) ◽  
pp. 20170419 ◽  
Author(s):  
Vadim Kaloshin ◽  
Alfonso Sorrentino
Keyword(s):  
Integrable Systems ◽  
Theme Issue ◽  
Finite Dimensional

In this survey, we provide a concise introduction to convex billiards and describe some recent results, obtained by the authors and collaborators, on the classification of integrable billiards, namely the so-called Birkhoff conjecture. This article is part of the theme issue ‘Finite dimensional integrable systems: new trends and methods’.

scholarly journals Positive-entropy integrable systems and the Toda lattice, II

2010 ◽  
Vol 149 (3) ◽  
pp. 491-538 ◽  
Author(s):  
LEO T. BUTLER
Keyword(s):  
Topological Entropy ◽  
Integrable Systems ◽  
Cotangent Bundle ◽  
Toda Lattice ◽  
Entropy Function ◽  
Topological Conjugacy ◽  
Lax Representation ◽  
Positive Entropy ◽  
Toral Automorphisms

AbstractThis paper constructs completely integrable convex Hamiltonians on the cotangent bundle of certain k bundles over l. A central role is played by the Lax representation of a Bogoyavlenskij–Toda lattice. The classification of these systems, up to iso-energetic topological conjugacy, is related to the classification of abelian groups of Anosov toral automorphisms by their topological entropy function.

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