Topology, singularities and integrability in Hamiltonian systems with two degrees of freedom

S. V. Bolotin ; V. V. Kozlov

Izvestiya. Mathematics , Tome 81 (2017) no. 4, pp. 671-687.

Résumé

We consider the problem of the existence of first integrals that are polynomial in momenta for Hamiltonian systems with two degrees of freedom on a fixed energy level (conditional Birkhoff integrals). It is assumed that the potential has several singular points. We show that in the presence of conditional polynomial integrals, the sum of degrees of the singularities does not exceed twice the Euler characteristic of the configuration space. The proof is based on introducing a complex structure on the configuration space and estimating the degree of the divisor corresponding to the leading term of the integral with respect to the momentum. We also prove that the topological entropy is positive under certain conditions.

Détail

Keywords: Hamiltonian system, integrability, singular point, regularization, Finsler metric
Mots-clés : conformal structure.

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