173 research outputs found

    Quadratic Volume-Preserving Maps: Invariant Circles and Bifurcations

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    We study the dynamics of the five-parameter quadratic family of volume-preserving diffeomorphisms of R^3. This family is the unfolded normal form for a bifurcation of a fixed point with a triple-one multiplier and also is the general form of a quadratic three-dimensional map with a quadratic inverse. Much of the nontrivial dynamics of this map occurs when its two fixed points are saddle-foci with intersecting two-dimensional stable and unstable manifolds that bound a spherical ``vortex-bubble''. We show that this occurs near a saddle-center-Neimark-Sacker (SCNS) bifurcation that also creates, at least in its normal form, an elliptic invariant circle. We develop a simple algorithm to accurately compute these elliptic invariant circles and their longitudinal and transverse rotation numbers and use it to study their bifurcations, classifying them by the resonances between the rotation numbers. In particular, rational values of the longitudinal rotation number are shown to give rise to a string of pearls that creates multiple copies of the original spherical structure for an iterate of the map.Comment: 53 pages, 29 figure

    Generalizations of the St\"ormer Problem for Dust Grain Orbits

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    We consider the generalized St\"ormer Problem that includes the electromagnetic and gravitational forces on a charged dust grain near a planet. For dust grains a typical charge to mass ratio is such that neither force can be neglected. Including the gravitational force gives rise to stable circular orbits that encircle that plane entirely above/below the equatorial plane. The effects of the different forces are discussed in detail. A modified 3rd Kepler's law is found and analyzed for dust grains.Comment: 21 pages LaTeX, 12 figure

    Resonances and Twist in Volume-Preserving Mappings

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    The phase space of an integrable, volume-preserving map with one action and dd angles is foliated by a one-parameter family of dd-dimensional invariant tori. Perturbations of such a system may lead to chaotic dynamics and transport. We show that near a rank-one, resonant torus these mappings can be reduced to volume-preserving "standard maps." These have twist only when the image of the frequency map crosses the resonance curve transversely. We show that these maps can be approximated---using averaging theory---by the usual area-preserving twist or nontwist standard maps. The twist condition appropriate for the volume-preserving setting is shown to be distinct from the nondegeneracy condition used in (volume-preserving) KAM theory.Comment: Many typos fixed and notation simplified. New nthn^{th} order averaging theorem and volume-preserving variant. Numerical comparison with averaging adde

    Nonexistence of an integral of the 6th degree in momenta for the Zipoy-Voorhees metric

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    We prove nonexistence of a nontrivial integral that is polynomial in momenta of degree less than 7 for the Zipoy-Voorhees spacetime with the parameter δ=2\delta=2Comment: 7 pages, no figure

    The Lie-Poisson structure of the reduced n-body problem

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    The classical n-body problem in d-dimensional space is invariant under the Galilean symmetry group. We reduce by this symmetry group using the method of polynomial invariants. As a result we obtain a reduced system with a Lie-Poisson structure which is isomorphic to sp(2n-2), independently of d. The reduction preserves the natural form of the Hamiltonian as a sum of kinetic energy that depends on velocities only and a potential that depends on positions only. Hence we proceed to construct a Poisson integrator for the reduced n-body problem using a splitting method.Comment: 26 pages, 2 figure

    Vanishing Twist near Focus-Focus Points

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    We show that near a focus-focus point in a Liouville integrable Hamiltonian system with two degrees of freedom lines of locally constant rotation number in the image of the energy-momentum map are spirals determined by the eigenvalue of the equilibrium. From this representation of the rotation number we derive that the twist condition for the isoenergetic KAM condition vanishes on a curve in the image of the energy-momentum map that is transversal to the line of constant energy. In contrast to this we also show that the frequency map is non-degenerate for every point in a neighborhood of a focus-focus point.Comment: 13 page

    A Poincar\'e section for the general heavy rigid body

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    A general recipe is developed for the study of rigid body dynamics in terms of Poincar\'e surfaces of section. A section condition is chosen which captures every trajectory on a given energy surface. The possible topological types of the corresponding surfaces of section are determined, and their 1:1 projection to a conveniently defined torus is proposed for graphical rendering.Comment: 25 pages, 10 figure

    Topology of energy surfaces and existence of transversal Poincar\'e sections

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    Two questions on the topology of compact energy surfaces of natural two degrees of freedom Hamiltonian systems in a magnetic field are discussed. We show that the topology of this 3-manifold (if it is not a unit tangent bundle) is uniquely determined by the Euler characteristic of the accessible region in configuration space. In this class of 3-manifolds for most cases there does not exist a transverse and complete Poincar\'e section. We show that there are topological obstacles for its existence such that only in the cases of S1×S2S^1\times S^2 and T3T^3 such a Poincar\'e section can exist.Comment: 10 pages, LaTe

    The problem of two fixed centres: bifurcations, actions, monodromy

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    A comprehensive analysis of the Euler-Jacobi problem of motion in the field of two fixed attracting centers is given, first classically and then quantum mechanically in semiclassical approximation. The system was originally studied in the context of celestial mechanics but, starting with Pauli’s dissertation, became a model for one-electron molecules such as H+ 2 (symmetric case of equal centers) or HHe2+ (asymmetric case of different centers). The present paper deals with arbitrary relative strength of the two centers and considers separately the planar and the three-dimensional problems. All versions represent nontrivial examples of integrable dynamics and are studied here from the unifying point of view of the energy momentum mapping from phase space to the space of integration constants. The interesting objects are the critical values of this mapping, i. e., its bifurcation diagram, and their pre-images which organize the foliation of phase space into Liouville-Arnold tori. The classical analysis culminates in the explicit derivation of the action variable representation of iso-energetic surfaces. The attempt to identify a system of global actions, smoothly dependent on the integration constants wherever these are non-critical, leads to the detection of monodromy of a special kind which is here described for the first time. The classical monodromy has its counterpart in the quantum version of the two-center problem where it prevents the assignments of unique quantum numbers even though the system is separable
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