35 research outputs found
Dynamics of the Tippe Top -- properties of numerical solutions versus the dynamical equations
We study the relationship between numerical solutions for inverting Tippe Top
and the structure of the dynamical equations. The numerical solutions confirm
oscillatory behaviour of the inclination angle for the symmetry
axis of the Tippe Top. They also reveal further fine features of the dynamics
of inverting solutions defining the time of inversion. These features are
partially understood on the basis of the underlying dynamical equations
Stationary problems for equation of the KdV type and dynamical -matrices.
We study a quite general family of dynamical -matrices for an auxiliary
loop algebra related to restricted flows for equations of
the KdV type. This underlying -matrix structure allows to reconstruct Lax
representations and to find variables of separation for a wide set of the
integrable natural Hamiltonian systems. As an example, we discuss the
Henon-Heiles system and a quartic system of two degrees of freedom in detail.Comment: 25pp, LaTe
Dynamics of the Tippe Top—Properties of numerical solutions versus the dynamical equations
Phase Space of Rolling Solutions of the Tippe Top
Equations of motion of an axially symmetric sphere rolling and sliding on a plane are usually taken as model of the tippe top. We study these equations in the nonsliding regime both in the vector notation and in the Euler angle variables when they admit three integrals of motion that are linear and quadratic in momenta. In the Euler angle variables (θ,φ,ψ) these integrals give separation equations that have the same structure as the equations of the Lagrange top. It makes it possible to describe the whole space of solutions by representing them in the space of parameters (D,λ,E) being constant values of the integrals of motion
The Lax pairs for the Holt system
By using non-canonical transformation between the Holt system and the
Henon-Heiles system the Lax pairs for all the integrable cases of the Holt
system are constructed from the known Lax representations for the Henon-Heiles
system.Comment: 7 pages, LaTeX2e, a4.st
Quasi-point separation of variables for the Henon-Heiles system and a system with quartic potential
We examine the problem of integrability of two-dimensional Hamiltonian
systems by means of separation of variables. The systematic approach to
construction of the special non-pure coordinate separation of variables for
certain natural two-dimensional Hamiltonians is presented. The relations with
SUSY quantum mechanics are discussed.Comment: 11 pages, Late
From bi-Hamiltonian geometry to separation of variables: stationary Harry-Dym and the KdV dressing chain
Separability theory of one-Casimir Poisson pencils, written down in arbitrary
coordinates, is presented. Separation of variables for stationary Harry-Dym and
the KdV dressing chain illustrates the theory.Comment: LaTex 14 pages, Proceedings of the Special Session on Integrable
Systems of the First Joint Meeting of the American Mathematical Society and
the Hong Kong Mathematical Society, to appear in J. Nonl. Math. Phy
Dynamics of the Tippe Top via Routhian Reduction
We consider a tippe top modeled as an eccentric sphere, spinning on a
horizontal table and subject to a sliding friction. Ignoring translational
effects, we show that the system is reducible using a Routhian reduction
technique. The reduced system is a two dimensional system of second order
differential equations, that allows an elegant and compact way to retrieve the
classification of tippe tops in six groups as proposed in [1] according to the
existence and stability type of the steady states.Comment: 16 pages, 7 figures, added reference. Typos corrected and a forgotten
term in de linearized system is adde
Separation of variables for the generalized Henon-Heiles system and system with quartic potential
We consider two well-known integrable systems on the plane using the concept
of natural Poisson bivectors on Riemaninan manifolds. Geometric approach to
construction of variables of separation and separated relations for the
generalized Henon-Heiles system and the generalized system with quartic
potential is discussed in detail.Comment: 9 pages, LaTeX with AMS font
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