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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 65th birthday
Abstract
Citation: Duistermaat J. J.,  Hanssmann H., SPECIAL ISSUE. Dedicated to Richard Cushman on the occasion of his 65th birthday, Regular and Chaotic Dynamics, 2007, vol. 12, no. 6, pp. 577-577
Fasso F.,  Ramos A.,  Sansonetto N.
On the Existence of Invariant Tori in Nearly-Integrable Hamiltonian Systems with Finitely Differentiable Perturbations
Abstract
We consider nonholonomic systems with linear, time-independent constraints subject to ositional conservative active forces. We identify a distribution on the configuration manifold, that we call the reaction-annihilator 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.
Keywords: nonholonomic systems, first integrals, first integrals linear in the velocities, symmetries of nonholonomic systems, reaction forces, Noether theorem
Citation: Fasso F.,  Ramos A.,  Sansonetto N., On the Existence of Invariant Tori in Nearly-Integrable Hamiltonian Systems with Finitely Differentiable Perturbations, Regular and Chaotic Dynamics, 2007, vol. 12, no. 6, pp. 579-588
DOI:10.1134/S1560354707060019
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.
Keywords: nonholonomic mechanics, variational principles, Lagrange–d'Alembert principle, contact order
Citation: Cuell C.,  Patrick G., Skew Critical Problems, Regular and Chaotic Dynamics, 2007, vol. 12, no. 6, pp. 589-601
DOI:10.1134/S1560354707060020
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.
Keywords: tippe top, eccentric sphere, Lagrangian equations, symmetries, Routhian reduction, relative equilibria, (linear) stability, bifurcation
Citation: Ciocci M.,  Langerock B., Dynamics of the Tippe Top via Routhian Reduction, Regular and Chaotic Dynamics, 2007, vol. 12, no. 6, pp. 602-614
DOI:10.1134/S1560354707060032
Sniatycki J.,  Cushman R.
Non-Holonomic Reduction of Symmetries, Constraints, and Integrability
Abstract
The aim of this paper is to give a brief description of singular reduction of non-holonomically constrained Hamiltonian systems.
Keywords: non-holonomic mechanics
Citation: Sniatycki J.,  Cushman R., Non-Holonomic Reduction of Symmetries, Constraints, and Integrability, Regular and Chaotic Dynamics, 2007, vol. 12, no. 6, pp. 615-621
DOI:10.1134/S1560354707060044
Horozov E.,  Stoyanova C.
Non-Integrability of Some Painlevé VI-Equations and Dilogarithms
Abstract
The paper studies the Painlevé VIe 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.
Keywords: integrability, Painlevé VI-equations, Hamiltonian system
Citation: Horozov E.,  Stoyanova C., Non-Integrability of Some Painlevé VI-Equations and Dilogarithms, Regular and Chaotic Dynamics, 2007, vol. 12, no. 6, pp. 622-629
DOI:10.1134/S1560354707060056
Beukers F.
Unitary Monodromy of Lamé Differential Operators
Abstract
The classical second order Lamé equation contains a so-called 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.
Keywords: unitary monodromy, Lamé differential equation
Citation: Beukers F., Unitary Monodromy of Lamé Differential Operators, Regular and Chaotic Dynamics, 2007, vol. 12, no. 6, pp. 630-641
DOI:10.1134/S1560354707060068
Cushman R.,  Dullin H. R.,  Hanssmann H.,  Schmidt D. S.
The $1:\pm2$ Resonance
Abstract
On the linear level elliptic equilibria of Hamiltonian systems are mere superpositions of harmonic oscillators. Non-linear 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 non-degenerate (i.e. the Kolmogorov non-degeneracy 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.
Keywords: resonant oscillators, normal form, singular reduction, bifurcation, energy-momentum mapping, monodromy
Citation: Cushman R.,  Dullin H. R.,  Hanssmann H.,  Schmidt D. S., The $1:\pm2$ Resonance, Regular and Chaotic Dynamics, 2007, vol. 12, no. 6, pp. 642-663
DOI:10.1134/S156035470706007X
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 4-DOF systems defined by perturbed isotropic oscillators (1-1-1-1 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.
Keywords: Hamiltonian system, bifurcation, normal form, reduction, Hamiltonian Hopf bifurcation, fourfold 1:1 resonance
Citation: Egea J.,  Ferrer S.,  van der Meer J., Hamiltonian Fourfold 1:1 Resonance with Two Rotational Symmetries, Regular and Chaotic Dynamics, 2007, vol. 12, no. 6, pp. 664-674
DOI:10.1134/S1560354707060081
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.
Keywords: monodromy, completely integrable systems
Citation: Bates L.,  Fasso F., An Affine Model for the Actions in an Integrable System with Monodromy, Regular and Chaotic Dynamics, 2007, vol. 12, no. 6, pp. 675-679
DOI:10.1134/S1560354707060093
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).
Keywords: integrable system, hyperbolic singularity, KAM theory, Kolmogorov condition
Citation: Zung N., Kolmogorov Condition near Hyperbolic Singularities of Integrable Hamiltonian Systems, Regular and Chaotic Dynamics, 2007, vol. 12, no. 6, pp. 680-688
DOI:10.1134/S156035470706010X
Dullin H. R.,  Vu-Ngoc S.
Symplectic Invariants Near Hyperbolic-Hyperbolic Points
Abstract
We construct symplectic invariants for Hamiltonian integrable systems of 2 degrees of freedom possessing a fixed point of hyperbolic-hyperbolic 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 energy-momentum map. This gives general conditions under which the non-degeneracy 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.
Keywords: completely integrable systems, hyperbolic-hyperbolic point, KAM, isoenergetic non-degeneracy, vanishing twist
Citation: Dullin H. R.,  Vu-Ngoc S., Symplectic Invariants Near Hyperbolic-Hyperbolic Points, Regular and Chaotic Dynamics, 2007, vol. 12, no. 6, pp. 689-716
DOI:10.1134/S1560354707060111
Giacobbe A.
Infinitesimally Stable and Unstable Singularities of 2-Degrees of Freedom Completely Integrable Systems
Abstract
In this article we give a list of 10 rank zero and 6 rank one singularities of 2-degrees of freedom completely integrable systems. Among such singularities, 14 are the singularities that satisfy a non-vanishing 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.
Keywords: singularities, completely integrable systems, bifurcation diagrams, infinitesimal deformations, cusps, local normal forms
Citation: Giacobbe A., Infinitesimally Stable and Unstable Singularities of 2-Degrees of Freedom Completely Integrable Systems, Regular and Chaotic Dynamics, 2007, vol. 12, no. 6, pp. 717-731
DOI:10.1134/S1560354707060123
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 3-dimensional irreducible blocks.
Keywords: joint covariant, quadratic, nilpotent normal form
Citation: Sanders  . A., Stanley Decomposition of the Joint Covariants of Three Quadratics, Regular and Chaotic Dynamics, 2007, vol. 12, no. 6, pp. 732-735
DOI:10.1134/S1560354707060135
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.
Keywords: integral medial axis, continued fraction
Citation: van der Kallen W., Integral Medial Axis and the Distance Between Closest Points, Regular and Chaotic Dynamics, 2007, vol. 12, no. 6, pp. 736-745
DOI:10.1134/S1560354707060147
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.
Keywords: odd symplectic group, adjoint orbit, distinguished type, type, modulus
Citation: Cushman R., Adjoint Orbits of the Odd Real Symplectic Group, Regular and Chaotic Dynamics, 2007, vol. 12, no. 6, pp. 746-755
DOI:10.1134/S1560354707060159

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