Topology, singularities and integrability in Hamiltonian systems with two degrees of freedom
Copyright policy )Topology, singularities and integrability in Hamiltonian systems with two degrees of freedom
Summary: 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.
- Steklov Mathematical Institute Russian Federation
- University of Wisconsin–Madison United States
- Department of Mathematical Sciences Russian Federation
Hamilton's equations, Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests, integrability, regularization, Relations of dynamical systems with symplectic geometry and topology, Ordinary differential equations and systems on manifolds, Hamiltonian system, singular point, conformal structure, Obstructions to integrability for finite-dimensional Hamiltonian and Lagrangian systems (nonintegrability criteria), Finsler metric
Hamilton's equations, Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests, integrability, regularization, Relations of dynamical systems with symplectic geometry and topology, Ordinary differential equations and systems on manifolds, Hamiltonian system, singular point, conformal structure, Obstructions to integrability for finite-dimensional Hamiltonian and Lagrangian systems (nonintegrability criteria), Finsler metric
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