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Liegroup methods
 ACTA NUMERICA
, 2000
"... Many differential equations of practical interest evolve on Lie groups or on manifolds acted upon by Lie groups. The retention of Liegroup structure under discretization is often vital in the recovery of qualitatively correct geometry and dynamics and in the minimization of numerical error. Having ..."
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Cited by 93 (18 self)
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Many differential equations of practical interest evolve on Lie groups or on manifolds acted upon by Lie groups. The retention of Liegroup structure under discretization is often vital in the recovery of qualitatively correct geometry and dynamics and in the minimization of numerical error. Having introduced requisite elements of differential geometry, this paper surveys the novel theory of numerical integrators that respect Liegroup structure, highlighting theory, algorithmic issues and a number of applications.
The Riemann Zeros and Eigenvalue Asymptotics
 SIAM Rev
, 1999
"... Comparison between formulae for the counting functions of the heights t n of the Riemann zeros and of semiclassical quantum eigenvalues En suggests that the t n are eigenvalues of an (unknown) hermitean operator H, obtained by quantizing a classical dynamical system with hamiltonian H cl . Many feat ..."
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Cited by 42 (5 self)
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Comparison between formulae for the counting functions of the heights t n of the Riemann zeros and of semiclassical quantum eigenvalues En suggests that the t n are eigenvalues of an (unknown) hermitean operator H, obtained by quantizing a classical dynamical system with hamiltonian H cl . Many features of H cl are provided by the analogy; for example, the "Riemann dynamics" should be chaotic and have periodic orbits whose periods are multiples of logarithms of prime numbers. Statistics of the t n have a similar structure to those of the semiclassical En ; in particular, they display randommatrix universality at short range, and nonuniversal behaviour over longer ranges. Very refined features of the statistics of the t n can be computed accurately from formulae with quantum analogues. The RiemannSiegel formula for the zeta function is described in detail. Its interpretation as a relation between long and short periodic orbits gives further insights into the quantum spectral fluctuations. We speculate that the Riemann dynamics is related to the trajectories generated by the classical hamiltonian H cl = XP. Key words. spectral asymptotics, number theory AMS subject classifications. 11M26, 11M06, 35P20, 35Q40, 41A60, 81Q10, 81Q50 PII. S0036144598347497 1.
Energy Surfaces of Ellipsoidal Billiards
 Z. Naturforsch
, 1996
"... Energy surfaces in the space of action variables are calculated and graphically presented for general triaxial ellipsoidal billiards. As was demonstrated by Jacobi in 1838, the system may be integrated in terms of hyperelliptic functions. The actual computation, however, has never been done. It is f ..."
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Cited by 4 (4 self)
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Energy surfaces in the space of action variables are calculated and graphically presented for general triaxial ellipsoidal billiards. As was demonstrated by Jacobi in 1838, the system may be integrated in terms of hyperelliptic functions. The actual computation, however, has never been done. It is found that generic energy surfaces consist of seven pieces, representing topologically different types of invariant tori. The character of the corresponding motion is discussed. Frequencies, winding numbers, and the location of resonances are also determined. The results may serve as a basis for perturbation theory of slightly modified systems, and for semiclassical quantization. 1 1 Introduction Hamiltonian systems with three degrees of freedom have recently attracted considerable attention. Even though the interest is primarily directed towards nonintegrable dynamics, it is important to have integrable limiting cases available as a reference. Ellipsoidal billiards are highly nontrivial...
Spherical Pendulum, Actions, and Spin
, 1996
"... The classical and quantum mechanics of a spherical pendulum are worked out, including the dynamics of a suspending frame with moment of inertia `. The presence of two separatrices in the bifurcation diagram of the energymomentum mapping has its mathematical expression in the hyperelliptic nature o ..."
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Cited by 3 (3 self)
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The classical and quantum mechanics of a spherical pendulum are worked out, including the dynamics of a suspending frame with moment of inertia `. The presence of two separatrices in the bifurcation diagram of the energymomentum mapping has its mathematical expression in the hyperelliptic nature of the problem. Nevertheless, numerical computation allows to obtain the action variable representation of energy surfaces, and to derive frequencies and winding ratios from there. The quantum mechanics is also best understood in terms of these actions. The limit ` ! 0 is of particular interest, both classically and quantum mechanically, as it generates two copies of the frameless standard spherical pendulum. This is suggested as a classical interpretation of spin. 1 Introduction John Ross was born in the year when Schrodinger's equation and Born's statistical interpretation of the wave function were published. The triumph of quantum theory left only minor roles for classical mechanics, in...
Chaos in Anisotropic PreInflationary Universes
, 2008
"... We study the dynamics of anisotropic Bianchi typeIX models with matter and cosmological constant. The models can be thought as describing the role of anisotropy in the early stages of inflation, where the cosmological constant Λ plays the role of the vacuum energy of the inflaton field. The concurr ..."
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Cited by 1 (0 self)
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We study the dynamics of anisotropic Bianchi typeIX models with matter and cosmological constant. The models can be thought as describing the role of anisotropy in the early stages of inflation, where the cosmological constant Λ plays the role of the vacuum energy of the inflaton field. The concurrence of the cosmological constant and anisotropy are sufficient to produce a chaotic dynamics in the gravitational degrees of freedom, connected to the presence of a critical point of saddlecenter type in the phase space of the system. In the neighborhood of the saddlecenter, the phase space presents the structure of cylinders emanating from unstable periodic orbits. The nonintegrability of the system implies that the extension of the cylinders away from this neighborhood has a complicated structure arising from their transversal crossings, resulting in a chaotic dynamics. The invariant character of chaos is guaranteed by the topology of cylinders. The model also presents a strong asymptotic de Sitter attractor but the way out from the initial singularity to the inflationary phase is completely chaotic. For a large set of initial conditions, even with very small anisotropy, the gravitational degrees of freedom oscillate a long time in the neighborhood of the saddlecenter before recollapsing or escaping to the de Sitter phase. These oscillations may provide a resonance mechanism for amplification of specific wavelengths of inhomogeneous fluctuations in the models. A geometrical interpretation is given for Wald’s inequality in terms of invariant tori and their destruction by increasing values of the cosmological constant. 1 1
1236 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 57 Balance and the Slow Quasimanifold: Some Explicit Results
, 1998
"... The ultimate limitations of the balance, slowmanifold, and potential vorticity inversion concepts are investigated. These limitations are associated with the weak but nonvanishing spontaneousadjustment emission, or Lighthill radiation, of inertia–gravity waves by unsteady, twodimensional or layer ..."
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The ultimate limitations of the balance, slowmanifold, and potential vorticity inversion concepts are investigated. These limitations are associated with the weak but nonvanishing spontaneousadjustment emission, or Lighthill radiation, of inertia–gravity waves by unsteady, twodimensional or layerwisetwodimensional vortical flow (the wave emission mechanism sometimes being called ‘‘geostrophic’ ’ adjustment even though it need not take the flow toward geostrophic balance). Spontaneousadjustment emission is studied in detail for the case of unbounded fplane shallowwater flow, in which the potential vorticity anomalies are confined to a finitesized region, but whose distribution within the region is otherwise completely general. The approach assumes that the Froude number F and Rossby number R satisfy F K 1 and R � 1 (implying, incidentally, that any balance would have to include gradient wind and other ageostrophic contributions). The method of matched asymptotic expansions is used to obtain a general mathematical description of spontaneousadjustment emission in this parameter regime. Expansions are carried out to O(F4), which is a high enough order to describe not only the weakly emitted waves but also, explicitly, the correspondingly weak radiation reaction upon the vortical flow, accounting for the loss of vortical energy. Exact evolution on a slow manifold, in its usual strict sense, would be incompatible with the arrow of time introduced by this radiation reaction and energy loss. The magnitude O(F4) of the radiation reaction may thus be taken to measure the degree of ‘‘fuzziness’ ’ of the entity that must exist in place of the strict slow manifold. That entity must, presumably, be not a simple invariant manifold, but rather an O(F4)thin, multileaved, fractal ‘‘stochastic layer’ ’ like those known for analogous but loworder coupled oscillator systems. It could more appropriately be called the ‘‘slow quasimanifold.’’ 1.
Planar resonant periodic orbits in Kuiper belt dynamics
, 2008
"... In the framework of the planar restricted three body problem we study a considerable number of resonances associated to the Kuiper Belt dynamics and located between 30 and 48 a.u. Our study is based on the computation of resonant periodic orbits and their stability. Stable periodic orbits are surrou ..."
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In the framework of the planar restricted three body problem we study a considerable number of resonances associated to the Kuiper Belt dynamics and located between 30 and 48 a.u. Our study is based on the computation of resonant periodic orbits and their stability. Stable periodic orbits are surrounded by regular librations in phase space and in such domains the capture of transNeptunian object is possible. All the periodic orbits found are symmetric and there is evidence for the existence of asymmetric ones only in few cases. In the present work first, second and third order resonances are under consideration. In the planar circular case we found that most of the periodic orbits are stable. The families of periodic orbits are temporarily interrupted by collisions but they continue up to relatively large values of the Jacobi constant and highly eccentric regular motion exists for all cases. In the elliptic problem and for a particular eccentricity value of the primary bodies the periodic orbits are isolated. The corresponding families, where they belong to, bifurcate from specific periodic orbits of the circular problem and seem to continue up to the rectilinear problem. Both stable and unstable orbits are obtained for each case. In the elliptic problem the unstable orbits found are associated with narrow chaotic domains in phase space. The evolution of the orbits, which are located in such chaotic domains, seems to be practically regular and bounded for long time intervals.
Gallium Nitride resonators / lasers. Directionality and Vector Resonances of Regular and Chaotic Dielectric
, 2004
"... Dielectric microcavities / microlasers are a key component for novel optoelectronic devices. We model such devices as a dielectric rod and analyze the vector waveequation for an infinite dielectric rod with arbitrary crosssection. Analytic results for the resonance condition and polarization prope ..."
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Dielectric microcavities / microlasers are a key component for novel optoelectronic devices. We model such devices as a dielectric rod and analyze the vector waveequation for an infinite dielectric rod with arbitrary crosssection. Analytic results for the resonance condition and polarization properties are given for the cylinder. With the parabolic equation method we derive the resonance condition and the polarization properties for modes related to 2d stable periodic ray orbits. The polarization of hybrid emitting modes of cylindrical resonators is shown to be linear up to a polarization critical angle and elliptical beyond this angle, which always lies between the Brewster and the totalinternalreflection angles of the dielectric. Arbitrary crosssections in general give rise to nonintegrable ray dynamics. We review classical Hamiltonian dynamics and billiards and expand the theory to dielectric billiards. Analysis of periodic ray orbit bifurcations and unstable manifolds enables us to understand the emission directionality of differently deformed polymer microlasers. We report new experiments on the first whispering gallery laser from a
Elliptic vortex patches: coasts and chaos
"... We investigate the interaction of ocean vortex patches with boundaries via the elliptic moment model of Melander, Zabusky & Stysek [13], under which vortex patches are approximated as elliptic regions of uniform vorticity. The interaction with a straight boundary is shown to reduce to the problem of ..."
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We investigate the interaction of ocean vortex patches with boundaries via the elliptic moment model of Melander, Zabusky & Stysek [13], under which vortex patches are approximated as elliptic regions of uniform vorticity. The interaction with a straight boundary is shown to reduce to the problem of an ellipse in constant strain, as previously investigated by Kida [9]. Interactions with more complicated geometries, where we might have expected the motion to be chaotic, are found to be constrained by an adiabatic invariant arising from a separation of time scales between the rotation of the patch and its motion along the coast. A couple of examples where the presence of a background flow does produce clear chaotic motion are also given. Finally a conformal mapping technique is developed to find the motion of vortex patches in geometries where the flow is not so easily calculated. 1