Three natural mechanical systems on Stiefel varieties

We consider integrable generalizations of the spherical pendulum system to the Stiefel variety $V(n,r)=SO(n)/SO(n-r)$ for a certain metric. For the case of V(n,2) an alternative integrable model of the pendulum is presented. We also describe a system on the Stiefel variety with a four-degree potential. The latter has invariant relations on $T^*V(n,r)$ which provide the complete integrability of the flow reduced on the oriented Grassmannian variety $G^+(n,r)=SO(n)/SO(r)\times SO(n-r)$.Comment: 14 page

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 $C^\infty$ functions and has positive topological entropy is constructed.Comment: 7 pages, LaTe

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

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

On integrable system on S2S^2 with the second integral quartic in the momenta

We consider integrable system on the sphere $S^2$ 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

On the stability problem for the so(5)\mathfrak{so}(5) free rigid body

In the general case of the $\mathfrak{so}(n)$ free rigid body we give a list of integrals of motion, which generate the set of Mishchenko's integrals. In the case of $\mathfrak{so}(5)$ we prove that there are fifteen coordinate type Cartan subalgebras which on a regular adjoint orbit give fifteen Weyl group orbits of equilibria. These coordinate type Cartan subalgebras are the analogues of the three axes of equilibria for the classical rigid body on $\mathfrak{so}(3)$. The nonlinear stability and instability of these equilibria is analyzed. In addition to these equilibria there are ten other continuous families of equilibria.Comment: Preprint of an article submitted for consideration in International Journal of Geometric Methods in Modern Physics \copyright 2011 copyright World Scientific Publishing Company http://www.worldscinet.com/ijgmmp

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 $S^1\times S^2$ and $T^3$ such a Poincar\'e section can exist.Comment: 10 pages, LaTe

Integrable matrix equations related to pairs of compatible associative algebras

We study associative multiplications in semi-simple associative algebras over C compatible with the usual one. An interesting class of such multiplications is related to the affine Dynkin diagrams of A, D, E-type. In this paper we investigate in details the multiplications of the A-type and integrable matrix ODEs and PDEs generated by them.Comment: 12 pages, Late