Volume 12, Number 6
Volume 12, Number 6, 2007
On the 65th birthday of R.Cushman
Duistermaat J. J., Hanßmann H.
SPECIAL ISSUE. Dedicated to Richard Cushman on the occasion of his 65^{th} birthday
Abstract

Fasso F., Ramos A., Sansonetto N.
On the Existence of Invariant Tori in NearlyIntegrable Hamiltonian Systems with Finitely Differentiable Perturbations
Abstract
We consider nonholonomic systems with linear, timeindependent constraints subject to ositional conservative active forces. We identify a distribution on the configuration manifold, that we call the reactionannihilator distribution $\mathcal{R}^\circ$, the fibers of which are the annihilators of the set of all values taken by the reaction forces on the fibers of the constraint distribution. We show that this distribution, which can be effectively computed in specific cases, plays a central role in the study of first integrals linear in the velocities of this class of nonholonomic systems. In particular we prove that, if the Lagrangian is invariant under (the lift of) a group action in the configuration manifold, then an infinitesimal generator of this action has a conserved momentum if and only if it is a section of the distribution $\mathcal{R}^\circ$. Since the fibers of $\mathcal{R}^\circ$ contain those of the constraint distribution, this version of the nonholonomic Noether theorem accounts for more conserved omenta than what was known so far. Some examples are given.

Cuell C., Patrick G.
Skew Critical Problems
Abstract
Skew critical problems occur in continuous and discrete nonholonomic Lagrangian systems. They are analogues of constrained optimization problems, where the objective is differentiated in directions given by an apriori distribution, instead of tangent directions to the constraint. We show semiglobal existence and uniqueness for nondegenerate skew critical problems, and show that the solutions of two skew critical problems have the same contact as the problems themselves. Also, we develop some infrastructure that is necessary to compute with contact order geometrically, directly on manifolds.

Ciocci M., Langerock B.
Dynamics of the Tippe Top via Routhian Reduction
Abstract
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 CBJB according to the existence and stability type of the steady states.

Sniatycki J., Cushman R.
NonHolonomic Reduction of Symmetries, Constraints, and Integrability
Abstract
The aim of this paper is to give a brief description of singular reduction of nonholonomically constrained Hamiltonian systems.

Horozov E., Stoyanova C.
NonIntegrability of Some Painlevé VIEquations and Dilogarithms
Abstract
The paper studies the Painlevé VI^{e} equations from the point of view of Hamiltonian nonintegrability.
For certain infinite number of points in the parameter space we prove that the equations
are not integrable. Our approach uses recent advance in Hamiltonian integrability reducing the
problem to higher differential Galois groups as well as the monodromy of dilogarithic functions.

Beukers F.
Unitary Monodromy of Lamé Differential Operators
Abstract
The classical second order Lamé equation contains a socalled accessory parameter $B$. In this paper we study for which values of $B$ the Lamé equation has a monodromy group which is conjugate to a subgroup of $SL(2, \mathbb{R})$ (unitary monodromy with indefinite hermitian form). We reformulate the problem as a spectral problem and give an asymptotic expansion for the spectrum.

Cushman R., Dullin H. R., Hanßmann H., Schmidt S.
The $1:\pm2$ Resonance
Abstract
On the linear level elliptic equilibria of Hamiltonian systems are mere superpositions of harmonic oscillators. Nonlinear terms can produce instability, if the ratio of frequencies is rational and the Hamiltonian is indefinite. In this paper we study the frequency ratio $\pm 1/2$ and its unfolding. In particular we show that for the indefinite case ($1:2$) the frequency ratio map in a neighborhood of the origin has a critical point, i.e. the twist condition is violated for one torus on every energy surface near the energy of the equilibrium. In contrast, we show that the frequency map itself is nondegenerate (i.e. the Kolmogorov nondegeneracy condition holds) for every torus in a neighborhood of the equilibrium point. As a by product of our analysis of the frequency map we obtain another proof of fractional monodromy in the $1:2$ resonance.

Egea J., Ferrer S., van der Meer J.
Hamiltonian Fourfold 1:1 Resonance with Two Rotational Symmetries
Abstract
In this communication we deal with the analysis of Hamiltonian Hopf bifurcations in 4DOF systems defined by perturbed isotropic oscillators (1111 resonance), in the presence of two quadratic symmetries $I_1$ and $I_2$. As a perturbation we consider a polynomial function with a parameter. After normalization, the truncated normal form gives rise to an integrable system which is analyzed using reduction to a one degree of freedom system. The Hamiltonian Hopf bifurcations are found using the 'geometric method' set up by one of the authors.

Bates L., Fasso F.
An Affine Model for the Actions in an Integrable System with Monodromy
Abstract
We give a standard model for the flat affine geometry defined by the local action variables of a completely integrable system. We are primarily interested in the affine structure in the neighborhood of a critical value with nontrivial monodromy.

Zung N.
Kolmogorov Condition near Hyperbolic Singularities of Integrable Hamiltonian Systems
Abstract
In this paper we show that, if an integrable Hamiltonian system admits a nondegenerate hyperbolic singularity then it will satisfy the Kolmogorov condegeneracy condition near that singularity (under a mild additional condition, which is trivial if the singularity contains a fixed point).

Dullin H. R., VuNgoc S.
Symplectic Invariants Near HyperbolicHyperbolic Points
Abstract
We construct symplectic invariants for Hamiltonian integrable systems of 2 degrees of freedom possessing a fixed point of hyperbolichyperbolic type. These invariants consist in some signs which determine the topology of the critical Lagrangian fibre, together with several Taylor series which can be computed from the dynamics of the system. We show how these series are related to the singular asymptotics of the action integrals at the critical value of the energymomentum map. This gives general conditions under which the nondegeneracy conditions arising in the KAM theorem (Kolmogorov condition, twist condition) are satisfied. Using this approach, we obtain new asymptotic formulae for the action integrals of the C.~Neumann system. As a corollary, we show that the Arnold twist condition holds for generic frequencies of this system.

Giacobbe A.
Infinitesimally Stable and Unstable Singularities of 2Degrees of Freedom Completely Integrable Systems
Abstract
In this article we give a list of 10 rank zero and 6 rank one singularities of 2degrees of freedom completely integrable systems. Among such singularities, 14 are the singularities that satisfy a nonvanishing condition on the quadratic part, the remaining 2 are rank 1 singularities that play a role in the geometry of completely integrable systems with fractional monodromy. We describe which of them are stable and which are unstable under infinitesimal completely integrable deformations of the system.

Sanders . A.
Stanley Decomposition of the Joint Covariants of Three Quadratics
Abstract
The Stanley decomposition of the joint covariants of three quadratics is computed using a new transvectant algorithm and computer algebra. This is sufficient to compute the general form of the normal form with respect to a nilpotent with three 3dimensional irreducible blocks.

van der Kallen W.
Integral Medial Axis and the Distance Between Closest Points
Abstract
In dimension two we prove an inequality that implies a desirable property of the integral medial axis as defined by Hesselink in Hesselink. In dimension three we conjecture a similar inequality.

Cushman R.
Adjoint Orbits of the Odd Real Symplectic Group
Abstract
In this paper we classify the adjoint orbits of the odd symplectic group over the field of real numbers. We need a noneigenvalue modulus to classify certain orbits.
