756 research outputs found
The quasi-bi-Hamiltonian formulation of the Lagrange top
Starting from the tri-Hamiltonian formulation of the Lagrange top in a
six-dimensional phase space, we discuss the possible reductions of the Poisson
tensors, the vector field and its Hamiltonian functions on a four-dimensional
space. We show that the vector field of the Lagrange top possesses, on the
reduced phase space, a quasi-bi-Hamiltonian formulation, which provides a set
of separation variables for the corresponding Hamilton-Jacobi equation.Comment: 12 pages, no figures, LaTeX, to appear in J. Phys. A: Math. Gen.
(March 2002
On Integrable Perturbations of Some Nonholonomic Systems
Integrable perturbations of the nonholonomic Suslov, Veselova, Chaplygin and
Heisenberg problems are discussed in the framework of the classical
Bertrand-Darboux method. We study the relations between the Bertrand-Darboux
type equations, well studied in the holonomic case, with their nonholonomic
counterparts and apply the results to the construction of nonholonomic
integrable potentials from the known potentials in the holonomic case
Lie Groups, Cluster Variables and Integrable Systems
We discuss the Poisson structures on Lie groups and propose an explicit
construction of the integrable models on their appropriate Poisson
submanifolds. The integrals of motion for the SL(N)-series are computed in
cluster variables via the Lax map. This construction, when generalised to the
co-extended loop groups, gives rise not only to several alternative
descriptions of relativistic Toda systems, but allows to formulate in general
terms some new class of integrable models.Comment: Based on talks given at Versatility of integrability, Columbia
University, May 2011; Simons Summer Workshop on Geometry and Physics, Stony
Brook, July-August 2011; Classical and Quantum Integrable Systems, Dubna,
January 2012; Progress in Quantum Field Theory and String Theory, Osaka,
April 2012; Workshop on Combinatorics of Moduli Spaces and Cluster Algebras,
Moscow, May-June 201
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