This article is cited in 13 papers

JÜRGEN MOSER – 80

Foliations of isonergy surfaces and singularities of curves

M. Radnović^a, V. Rom-Kedar<sup>b

a</sup> Mathematical Institute SANU, Belgrade, Serbia
^b The Estrin Family Chair of Computer Science and Applied Mathematics, The Weizmann Institute of Science, Rehovot 76100, Israel

Abstract: It is well known that changes in the Liouville foliations of the isoenergy surfaces of an integrable system imply that the bifurcation set has singularities at the corresponding energy level.We formulate certain genericity assumptions for two degrees of freedom integrable systems and we prove the opposite statement: the essential critical points of the bifurcation set appear only if the Liouville foliations of the isoenergy surfaces change at the corresponding energy levels. Along the proof, we give full classification of the structure of the isoenergy surfaces near the critical set under our genericity assumptions and we give their complete list using Fomenko graphs. This may be viewed as a step towards completing the Smale program for relating the energy surfaces foliation structure to singularities of the momentum mappings for non-degenerate integrable two degrees of freedom systems.

Keywords: Hamiltonian system, integrable system, singularity, Liouville foliation, isoenergy manifold, bifurcation set, Liouville equivalence.

MSC: 37J35, 70H06, 37C15, 70H08

Received: 05.10.2008
Accepted: 06.11.2008

Language: English

DOI: 10.1134/S1560354708060117

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