Volume 11, Number 1
Volume 11, Number 1, 2006
Benettin G., Fasso F., Guzzo M.
Long term stability of proper rotations and local chaotic motions in the perturbed Euler rigid body
Abstract
The long term stability of the proper rotations of the perturbed Euler rigid body was recently investigated analytically in the framework of Nekhoroshev theory. In this paper we perform a parallel numerical investigation, with the double aim of illustrating the theory and to submit it to a critical test. We focus the attention on the case of resonant motions, for which the stability is not trivial (resonant proper rotations are indeed stable in spite of the presence of local chaotic motions, with positive Lyapunov exponent, around them). The numerical results indicate that the analytic results are essentially optimal, apart from a particular resonance, actually the lowest order one, where the system turns out to be more stable than the theoretical expectation.

Jalnine A. Y., Kuznetsov S. P., Osbaldestin A. H.
Dynamics of small perturbations of orbits on a torus in a quasiperiodically forced 2D dissipative map
Abstract
We consider the dynamics of small perturbations of stable twofrequency quasiperiodic orbits on an attracting torus in the quasiperiodically forced Hénon map. Such dynamics consists in an exponential decay of the radial component and in a complex behaviour of the angle component. This behaviour may be two or threefrequency quasiperiodicity, or it may be irregular. In the latter case a graphic image of the dynamics of the perturbation angle is a fractal object, namely a strange nonchaotic attractor, which appears in auxiliary map for the angle component. Therefore, we claim that stable trajectories may approach the attracting torus either in a regular or in an irregular way. We show that the transition from quasiperiodic dynamics to chaos in the model system is preceded by the appearance of an irregular behaviour in the approach of the perturbed quasiperiodic trajectories to the smooth attracting torus. We also demonstrate a link between the evolution operator of the perturbation angle and a quasiperiodically forced circle mapping of a special form and with a Harper equation with quasiperiodic potential.

Cushman R., Duistermaat J. J.
Nearly flat falling motions of the rolling disk
Abstract
We study the motion of a disk which rolls on a horizontal plane under the influence of gravity, without slipping or loss of energy due to friction. There is a codimension one semianalytic subset $F$ of the phase space such that the disk falls flat in a finite time, if and only if its initial phase point belongs to $F$. We describe the motion of the disk when it starts at a point $p \notin F$ which is close to a point $f \in F$. It then almost falls flat, after which it rises up again. We prove that during the short time interval that the disk is almost flat, the point of contact races around the rim of the disk from a well defined position at the end of falling to a well defined position at the beginning of rising, where the increase of the angle only depends on the mass distribution of the disk and the radius of the rim. The sign of the increase of the angle depends on the side of $F$ from which $p$ approaches $f$.

Gorelyshev I. V.
On the full number of collisions in certain onedimensional billiard problems
Abstract
In the present work we consider motion of a light particle between a wall and a massive particle. Collisions in the system are elastic. In [1] the full number of collisions in this system was calculated. It turned out to be approximately equal to the product of number $\pi$ and the square root of ratio of the particles' masses. This formula was derived using reduction of the system to a billiard. In the present work this result is derived by means of the adiabatic perturbation theory for systems with impacts [2].

Cendra H., Diaz V. A.
The Lagrange–D'Alembert–Poincaré equations and integrability for the rolling disk
Abstract
Classical nonholonomic systems are described by the Lagrange–d'Alembert principle. The presence of symmetry leads, upon the choice of an arbitrary principal connection, to a reduced variational principle and to the Lagrange–d'Alembert–Poincaré reduced equations. The case of rolling bodies has a long history and it has been the purpose of many works in recent times, in part because of its applications to robotics. In this paper we study the classical example of the rolling disk. We consider a natural abelian group of symmetry and a natural connection for this example and obtain the corresponding Lagrange–d'Alembert–Poincaré equations written in terms of natural reduced variables. One interesting feature of this reduced equations is that they can be easily transformed into a single ordinary equation of second order, which is a Heun's equation.

Rudnev M., Ten V. V.
A model for separatrix splitting near multiple resonances
Abstract
We propose and study a model for local dynamics of a perturbed convex realanalytic Liouvilleintegrable Hamiltonian system near a resonance of multiplicity $1+m$, $m \geqslant 0$. Physically, the model represents a toroidal pendulum, coupled with a Liouvilleintegrable system of $n$ nonlinear rotators via a small analytic potential. The global bifurcation problem is setup for the $n+1$ dimensional isotropic manifold, corresponding to a specific homoclinic orbit of the toroidal pendulum. The splitting of this manifold can be described by a scalar function on the $n$torus. A sharp estimate for its Fourier coefficients is proven. It generalizes to a multiple resonance normal form of a convex analytic Liouville nearintegrable Hamiltonian system. The bound then is exponentially small.

Yehia H. M.
Twodimensional conservative mechanical systems with quartic second integral
Abstract
The method introduced in [20] was applied in [21] and [22] for constructing integrable conservative two dimentional mechanical systems whose second integral of motion is polynomial up to third degree in the velocities. In this paper we apply the same method for systemaic construction of mechanical systems with a quartic integral. As in our previous works, the configuration space is not assumed an Euclidean plane. This widens the range of applicability of the results to diverse mechanical systems such as rigid body dynamics and motion on two dimensional surfaces of positive, negative and variable curvature. Two new severalparameter integrable systems are obtained, which unify and generalize several previously known ones by modifying the configuration manifold and the potential of the forces acting on the system. Those systems are shown to include as special cases, integrable problems of motion in the Euclidean plane, the hyperbolic plane and different types of curved two dimensional manifolds. The results are applied to problems of rigid body dynamics. New integrable cases are obtained as special versions of one of the new systems, corresponding to different choices of the parameters. Those cases include new generalizations of the classical cases of Kovalevskaya, Chaplygin and Goryachev.

Tronin K. G.
Absolute choreographies of point vortices on a sphere
Abstract
New periodic solutions to the problem of three and four identical vortices on a sphere are specified. These solutions correspond to choreographies of vortices on a plane. We also describe different methods to construct choreographies for an arbitrary number of vortices. The most interesting methods are based on splitting of the static symmetrical vortex configurations on a sphere.

Meletlidou E., Stagika G.
On the continuation of degenerate periodic orbits in Hamiltonian systems
Abstract
The continuation of nonisolated periodic orbits lying on the resonant invariant tori of an integrable Hamiltonian system with respect to a small perturbative parameter cannot be proved by a direct application of the continuation theorem, since their monodromy matrix possesses more than a single pair of unit eigenvalues. In this case one may use Poincaré's theorem which proves that, if the integrable part of the Hamiltonian is nondegenerate and the average value of the perturbing function, evaluated along the unperturbed periodic orbits, possesses a simple extremum on such an orbit, then this orbit can be analytically continued with respect to the perturbation. In the present paper we prove a criterion for the continuation of the nonisolated periodic orbits, for which this average value is constant along the periodic orbits of the resonant torus and Poincaré's theorem is not applicable. We apply the results in two such systems of two degrees of freedom.
