43 research outputs found
Topology of energy surfaces and existence of transversal Poincar\'e sections
Two questions on the topology of compact energy surfaces of natural two
degrees of freedom Hamiltonian systems in a magnetic field are discussed. We
show that the topology of this 3-manifold (if it is not a unit tangent bundle)
is uniquely determined by the Euler characteristic of the accessible region in
configuration space. In this class of 3-manifolds for most cases there does not
exist a transverse and complete Poincar\'e section. We show that there are
topological obstacles for its existence such that only in the cases of
and such a Poincar\'e section can exist.Comment: 10 pages, LaTe
Geometry of integrable dynamical systems on 2-dimensional surfaces
This paper is devoted to the problem of classification, up to smooth
isomorphisms or up to orbital equivalence, of smooth integrable vector fields
on 2-dimensional surfaces, under some nondegeneracy conditions. The main
continuous invariants involved in this classification are the left equivalence
classes of period or monodromy functions, and the cohomology classes of period
cocycles, which can be expressed in terms of Puiseux series. We also study the
problem of Hamiltonianization of these integrable vector fields by a compatible
symplectic or Poisson structure.Comment: 31 pages, 12 figures, submitted to a special issue of Acta
Mathematica Vietnamic
Foliations of Isonergy Surfaces and Singularities of Curves
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.Comment: 30 pages, 19 figure