174 research outputs found
Quadratic Volume-Preserving Maps: Invariant Circles and Bifurcations
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
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
The phase space of an integrable, volume-preserving map with one action and
angles is foliated by a one-parameter family of -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 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
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
Comment: 7 pages, no figure
The Lie-Poisson structure of the reduced n-body problem
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
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
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
The problem of two fixed centres: bifurcations, actions, monodromy
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
Topology of energy surfaces and existence of transversal Poincar\'e sections
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
and such a Poincar\'e section can exist.Comment: 10 pages, LaTe
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