11 research outputs found
Quasi-BiHamiltonian Systems and Separability
Two quasi--biHamiltonian systems with three and four degrees of freedom are
presented. These systems are shown to be separable in terms of Nijenhuis
coordinates. Moreover the most general Pfaffian quasi-biHamiltonian system with
an arbitrary number of degrees of freedom is constructed (in terms of Nijenhuis
coordinates) and its separability is proved.Comment: 10 pages, AMS-LaTeX 1.1, to appear in J. Phys. A: Math. Gen. (May
1997
Completeness of the cubic and quartic H\'enon-Heiles Hamiltonians
The quartic H\'enon-Heiles Hamiltonian passes the Painlev\'e test for
only four sets of values of the constants. Only one of these, identical to the
traveling wave reduction of the Manakov system, has been explicitly integrated
(Wojciechowski, 1985), while the three others are not yet integrated in the
generic case . We integrate them by building
a birational transformation to two fourth order first degree equations in the
classification (Cosgrove, 2000) of such polynomial equations which possess the
Painlev\'e property. This transformation involves the stationary reduction of
various partial differential equations (PDEs). The result is the same as for
the three cubic H\'enon-Heiles Hamiltonians, namely, in all four quartic cases,
a general solution which is meromorphic and hyperelliptic with genus two. As a
consequence, no additional autonomous term can be added to either the cubic or
the quartic Hamiltonians without destroying the Painlev\'e integrability
(completeness property).Comment: 10 pages, To appear, Theor.Math.Phys. Gallipoli, 34 June--3 July 200
On bi-integrable natural Hamiltonian systems on the Riemannian manifolds
We introduce the concept of natural Poisson bivectors, which generalizes the
Benenti approach to construction of natural integrable systems on the
Riemannian manifolds and allows us to consider almost the whole known zoo of
integrable systems in framework of bi-hamiltonian geometry.Comment: 24 pages, LaTeX with AMSfonts (some new references were added
The nonabelian Liouville-Arnold integrability by quadratures problem: a symplectic approach
A symplectic theory approach is devised for solving the problem of
algebraic-analytical construction of integral submanifold imbeddings for
integrable (via the nonabelian Liouville-Arnold theorem) Hamiltonian systems on
canonically symplectic phase spaces
A non-linear Oscillator with quasi-Harmonic behaviour: two- and -dimensional Oscillators
A nonlinear two-dimensional system is studied by making use of both the
Lagrangian and the Hamiltonian formalisms. The present model is obtained as a
two-dimensional version of a one-dimensional oscillator previously studied at
the classical and also at the quantum level. First, it is proved that it is a
super-integrable system, and then the nonlinear equations are solved and the
solutions are explicitly obtained. All the bounded motions are quasiperiodic
oscillations and the unbounded (scattering) motions are represented by
hyperbolic functions. In the second part the system is generalized to the case
of degrees of freedom. Finally, the relation of this nonlinear system with
the harmonic oscillator on spaces of constant curvature, two-dimensional sphere
and hyperbolic plane , is discussed.Comment: 30 pages, 4 figures, submitted to Nonlinearit