Topological and Geometrical Obstructions to Complete Integrability

Part of the Ergebnisse der Mathematik und ihrer Grenzgebiete book series (MATHE3, volume 31)


We start the analysis of the reasons for nonintegrable behavior of Hamiltonian systems with a discussion of relatively recently discovered “rough” topological obstructions to integrability. In [130] it was proved that a closed analytic surface with genus ≥ 2 cannot be the configuration space of an integrable analytic system. The reason is the existence of an infinite number of unstable periodic orbits on which the integrals are dependent. This result (unnoticed earlier due to the preference for local study of dynamical systems) was generalized in several directions. The proof of nonintegrability is based on variational methods and subtle results from the theory of singularities of analytic mappings.


Hamiltonian System Configuration Space Topological Entropy Closed Geodesic Complete Integrability 
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Copyright information

© Springer-Verlag Berlin Heidelberg 1996

Authors and Affiliations

  1. 1.Department of MathematicsMoscow State UniversityMoscowRussia

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