Geodesic
Topology, singularities and integrability in Hamiltonian systems with two degrees of freedom
Izvestiya. Mathematics , Tome 81 (2017) no. 4, pp. 671-687

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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.
Keywords: Hamiltonian system, integrability, singular point, regularization, Finsler metric
Mots-clés : conformal structure. 
@article{IM2_2017_81_4_a0,
     author = {S. V. Bolotin and V. V. Kozlov},
     title = {Topology, singularities and integrability in {Hamiltonian} systems with two degrees of freedom},
     journal = {Izvestiya. Mathematics },
     pages = {671--687},
     publisher = {mathdoc},
     volume = {81},
     number = {4},
     year = {2017},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2017_81_4_a0/}
}
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S. V. Bolotin; V. V. Kozlov. Topology, singularities and integrability in Hamiltonian systems with two degrees of freedom. Izvestiya. Mathematics , Tome 81 (2017) no. 4, pp. 671-687. http://geodesic.mathdoc.fr/item/IM2_2017_81_4_a0/