Existence of AN Additional First Integral and Completeness of the Flow for Hamiltonian Vector Fields

Details Existence of AN Additional First Integral and Completeness of the Flow for Hamiltonian Vector Fields Existence of AN Additional First Integral and Completeness of the Flow for Hamiltonian Vector Fields

Oct 2000

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2000hscm.conf..261m&link_type=abstract

HAMILTONIAN SYSTEMS AND CELESTIAL MECHANICS (HAMSYS-98). Proceedings of the III International Symposium. Held 7-11 December 1998

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Pairs of real analytic Hamiltonian vector fields Xh, Xg in Poisson involution over (not necessarily compact) symplectic manifolds are considered. We address the following problem: describe how a two-dimensional orbit {L} of the induced (R2, +)-action falls to an isolated common zero of Xh and Xg. A generalization of the Poincaré-Hopf index is introduced to describe the dynamics of Xh on {L}. Poincaré-Hopf index at least three on some {L}, implies that Xh), has incomplete flow (i.e. is not well defined for all time). Completeness of the Xh-flow implies Poincaré-Hopf index one or two on {L}, and a full description of {L} and Xh is provided. Explicit examples of the index computation are given.

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