180 research outputs found
Integrable geodesic flow with positive topological entropy
An example of a real-analytic metric on a compact manifold whose geodesic
flow is Liouville integrable by functions and has positive
topological entropy is constructed.Comment: 7 pages, LaTe
Topological monodromy as an obstruction to Hamiltonization of nonholonomic systems: pro or contra?
The phenomenon of a topological monodromy in integrable Hamiltonian and
nonholonomic systems is discussed. An efficient method for computing and
visualizing the monodromy is developed. The comparative analysis of the
topological monodromy is given for the rolling ellipsoid of revolution problem
in two cases, namely, on a smooth and on a rough plane. The first of these
systems is Hamiltonian, the second is nonholonomic. We show that, from the
viewpoint of monodromy, there is no difference between the two systems, and
thus disprove the conjecture by Cushman and Duistermaat stating that the
topological monodromy gives a topological obstruction for Hamiltonization of
the rolling ellipsoid of revolution on a rough plane.Comment: 31 pages, 11 figure
Integrable magnetic geodesic flows on Lie groups
Right-invariant geodesic flows on manifolds of Lie groups associated with
2-cocycles of corresponding Lie algebras are discussed. Algebra of integrals of
motion for magnetic geodesic flows is considered and necessary and sufficient
condition of integrability in quadratures is formulated. Canonic forms for
2-cocycles of all 4-dimensional Lie algebras are given and integrable cases
among them are separated.Comment: 16 page
Compatible Lie brackets related to elliptic curve
For the direct sum of several copies of sl_n, a family of Lie brackets
compatible with the initial one is constructed. The structure constants of
these brackets are expressed in terms of theta-functions associated with an
elliptic curve. The structure of Casimir elements for these brackets is
investigated. A generalization of this construction to the case of
vector-valued theta-functions is presented. The brackets define a
multi-hamiltonian structure for the elliptic sl_n-Gaudin model. A different
procedure for constructing compatible Lie brackets based on the argument shift
method for quadratic Poisson brackets is discussed.Comment: 18 pages, Late
On integrable system on with the second integral quartic in the momenta
We consider integrable system on the sphere with an additional integral
of fourth order in the momenta. At the special values of parameters this system
coincides with the Kowalevski-Goryachev-Chaplygin system.Comment: LaTeX, 6 page
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
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