72 research outputs found
On bi-hamiltonian structure of some integrable systems on so*(4)
We classify all the quadratic Poisson structures on and ,
which have the same foliation by symplectic leaves as the canonical Lie-Poisson
tensors. The separated variables for the some of the corresponding
bi-integrable systems are constructed.Comment: LaTeX with Amsfonts, 13 pages, corrected typo
On the Routh sphere problem
We discuss an embedding of a vector field for the nonholonomic Routh sphere
into a subgroup of commuting Hamiltonian vector fields on six dimensional phase
space. The corresponding Poisson brackets are reduced to the canonical Poisson
brackets on the Lie algebra e(3). It allows us to relate nonholonomic Routh
system with the Hamiltonian system on cotangent bundle to the sphere with
canonical Poisson structure.Comment: LaTeX with AMSFonts, 11 page
Addition theorems and the Drach superintegrable systems
We propose new construction of the polynomial integrals of motion related to
the addition theorems. As an example we reconstruct Drach systems and get some
new two-dimensional superintegrable Stackel systems with third, fifth and
seventh order integrals of motion.Comment: 18 pages, the talk given on the conference "Superintegrable Systems
in Classical and Quantum Mechanics", Prague 200
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
Duality between integrable Stackel systems
For the Stackel family of the integrable systems a non-canonical
transformation of the time variable is considered. This transformation may be
associated to the ambiguity of the Abel map on the corresponding hyperelliptic
curve. For some Stackel's systems with two degrees of freedom the 2x2 Lax
representations and the dynamical r-matrix algebras are constructed. As an
examples the Henon-Heiles systems, integrable Holt potentials and the
integrable deformations of the Kepler problem are discussed in detail.Comment: LaTeX2e, 18 page
The Maupertuis principle and canonical transformations of the extended phase space
We discuss some special classes of canonical transformations of the extended
phase space, which relate integrable systems with a common Lagrangian
submanifold. Various parametric forms of trajectories are associated with
different integrals of motion, Lax equations, separated variables and
action-angles variables. In this review we will discuss namely these induced
transformations instead of the various parametric form of the geometric
objects
Integrable Euler top and nonholonomic Chaplygin ball
We discuss the Poisson structures, Lax matrices, -matrices, bi-hamiltonian
structures, the variables of separation and other attributes of the modern
theory of dynamical systems in application to the integrable Euler top and to
the nonholonomic Chaplygin ball.Comment: 25 pages, LaTeX with AMS fonts, final versio
Canonical transformations of the extended phase space, Toda lattices and Stackel family of integrable systems
We consider compositions of the transformations of the time variable and
canonical transformations of the other coordinates, which map completely
integrable system into other completely integrable system. Change of the time
gives rise to transformations of the integrals of motion and the Lax pairs,
transformations of the corresponding spectral curves and R-matrices. As an
example, we consider canonical transformations of the extended phase space for
the Toda lattices and the Stackel systems.Comment: LaTeX2e + Amssymb, 22p
Canonical transformations of the time for the Toda lattice and the Holt system
For the Toda lattice and the Holt system we consider properties of canonical
transformations of the extended phase space, which preserve integrability. The
separated variables are invariant under change of the time. On the other hand,
mapping of the time induces transformations of the action-angles variables and
a shift of the generating function of the B\"{a}cklund transformation.Comment: LaTeX2e, +amssymb.cls, 8
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