Asymptotic Control Theory for a Closed String

We develop an asymptotical control theory for one of the simplest distributed oscillating systems, namely, for a closed string under a bounded load applied to a single distinguished point. We find exact classes of string states that admit complete damping and an asymptotically exact value of the required time. By using approximate reachable sets instead of exact ones, we design a dry-friction like feedback control, which turns out to be asymptotically optimal. We prove the existence of motion under the control using a rather explicit solution of a nonlinear wave equation. Remarkably, the solution is determined via purely algebraic operations. The main result is a proof of asymptotic optimality of the control thus constructed.     

Spinning Strings,Black Holes and Stable Closed Timelike Geodesics

The existence and stability under linear perturbation of closed timelike curves in the spacetime associated to Schwarzschild black hole pierced by a spinning string are studied. Due to the superposition of the black hole, we find that the spinning string spacetime is deformed in such a way to allow the existence of closed timelike geodesics.     

Closed Geodesics and Billiards on Quadrics Related to Elliptic KdV Solutions

We consider algebraic geometrical properties of the integrable billiard on a quadric Q with elastic impacts along another quadric confocal to Q. These properties are in sharp contrast with those of the ellipsoidal Birkhoff billiards in ⁿ . Namely, generic complex invariant manifolds are not Abelian varieties, and the billiard map is no more algebraic. A Poncelet-like theorem for such system is known. We give explicit sufficient conditions both for closed geodesics and periodic billiard orbits on Q and discuss their relation with the elliptic KdV solutions and elliptic Calogero system.     

The Correlation Between Multiplicities¶of Closed Geodesics on the Modular Surface

An explicit product representation is proved for the correlation function of the multiplicities of closed geodesics on the modular surface. This makes rigorous part of the investigation of Bogomolny, Leyvraz and Schmit on the correlation of the eigenvalues of the Laplacian on the modular surface. The result can also be seen as a (rigorously proved) analogue of the Hardy-Littlewood twin prime conjecture. Received: 25 May 2001 / Accepted: 28 August 2001     

A Closed Formula for the Asymptotic Expansion of the Bergman Kernel

We prove a graph theoretic closed formula for coefficients in the Tian-Yau-Zelditch asymptotic expansion of the Bergman kernel. The formula is expressed in terms of the characteristic polynomial of the directed graphs representing Weyl invariants. The proof relies on a combinatorial interpretation of a recursive formula due to M. Engliš and A. Loi.     

Asymptotic Dynamics of Self-driven Vehicles in a Closed Boundary

We study the asymptotic dynamics of self-driven vehicles in a loop using a car-following model with the consideration of volume exclusions. In particular, we derive the dynamical steady states for the single-cluster case and obtain the corresponding fundamental diagrams, exhibiting two branches representative of entering and leaving the jam, respectively. By simulations we find that the speed average over all vehicles eventually reaches the same value, regardless of final clustering states. The autocorrelation functions for overall speed average and single-vehicle speed are studied, each revealing a unique time scale. We also discuss the role of noises in vehicular accelerations. Based on our observations we give trial definitions about the degree of chaoticity for general self-driven many-body systems.     

Geodesics around oscillatons

Oscillatons are spherically symmetric solutions to the Einstein–Klein–Gordon equations. These solutions are non-singular, asymptotically flat, and with periodic time dependency. In this paper, we investigate the geodesic motion of particles moving around of an oscillatonic field. Bound orbits are found for particular values of the particles' angular momentum L and their initial radial position r ₀. It is found that the radial coordinate of such particles oscillates in time and we are able to predict the corresponding oscillating period as well as its amplitude. We carry out this study for the quadratic V(φ) = m _Φ Φ²/2 scalar field potential. We discuss possible ways to follow in order to connect this kind of studies with astrophysical observations.     

Geodesics in open universes

We present the geodesics on homogeneous and isotropic negatively curved spaces in a simple form suitable for application to cosmological problems. The pattern of geodesics translates into a pattern on the microwave background radiation. Generalizing, we discuss how the patterns in the microwave sky of anisotropic homogeneous universes can be predicted qualitatively by looking at the invariances that generate their three-geometries and their geodesics.     

Geodesics on loop spaces

The space of loops smoothly embedded into a Riemannian manifold, being a principal fibre bundle with structure group Diff S¹, is investigated from a Kaluza-Klein type point of view. In particular, the Levi-Civita connection for the natural Diff S¹-invariant metric on this loop space is calculated and the corresponding horizontal geodesics (the analogue of classical free motion of point particles) are characterized. Finally, an explicit solution is given in the case of loops in ³.     

Geodesics in rotating systems

The paper investigates the equations for geodesics, null geodesics, and spatial geodesics in rotating systems. Geodesics and null geodesics have, as usual, been interpreted as the paths of free particles and of light rays, respectively. Spatial geodesics are given a firm interpretation as the shortest paths between points within the rotating system, where the path length is measured by an observer in the rotating system who moves along the spatial geodesic. The paper shows that equations for geodesics in rotating systems may be derived by the traditional method, i.e., from the flat-space metric of general relativity, or by means of the instantaneous Lorentz frames approach. This supports the use of instantaneous Lorentz frames as a valid method for the analysis of events in rotating systems.     

Geodesics for the NUT metric and gravitational monopoles

In order to provide insight about the physical interpretation of the NUT parameter, we solve the geodesic equations for the NUT metric. We show that the properties of NUT geodesics are similar to the properties of trajectories for charged particles orbiting about a magnetic monopole. In summary, we show that (1) the orbits lie on the surface of a cone, (2) the conserved total angular momentum is the sum of the orbital angular momentum plus the angular momentum due to the monopole field, (3) the monopole field angular momentum is independent of the separation between the source of the gravitational field and the test particle, and (4) the geodesics are almost spherically symmetric. The strong similarities between the NUT geodesics and the electromagnetic monopole suggest that the NUT metric is an exact solution for a gravitational magnetic monopole. However, the subtle difference of being only almost spherically symmetric implies that the analogy is not perfect. The almost spherically symmetric nature of the NUT geodesics suggest that the energy of the Dirac string makes a contribution to the solution. We also construct exact solutions for special orbits, discuss a twin paradox, and speculate about the Dirac quantization condition for a gravitational magnetic monopole.     

Null Geodesics in Five-Dimensional Manifolds

We analyze a class of 5D non-compact warped-product spaces characterized by metrics that depend on the extra coordinate via a conformal factor. Our model is closely related to the so-called canonical coordinate gauge of Mashhoon et al. We confirm that if the 5D manifold in our model is Ricci-flat, then there is an induced cosmological constant in the 4D sub-manifold. We derive the general form of the 5D Killing vectors and relate them to the 4D Killing vectors of the embedded spacetime. We then study the 5D null geodesic paths and show that the 4D part of the motion can be timelike—that is, massless particles in 5D can be massive in 4D. We find that if the null trajectories are affinely parameterized in 5D, then the particle is subject to an anomalous acceleration or fifth force. However, this force may be removed by reparameterization, which brings the correct definition of the proper time into question. Physical properties of the geodesics—such as rest mass variations induced by a variable cosmological "constant," constants of the motion and 5D time-dilation effects—are discussed and are shown to be open to experimental or observational investigation.     

Geodesics in black hole space-times

We give the equations governing trajectories in black hole space-times, and present a guide to the literature-a who did what annotated reference list.     

Geodesics of Spherical Dilaton Spacetimes

The properties of spherical dilaton black hole spacetimes are investigated through a study of their geodesics. The closed and non-closed orbits of test particles are analysed using the effective potential and phase-plane method. The stability and types of orbits are determined in terms of the energy and angular momentum of the test particles. The conditions of the existence of circular orbits for a spherical dilaton spacetime with an arbitrary dilaton coupling constant α are obtained. The properties of the orbits and in particular the position of the innermost stable circular orbit are compared to those of the Reissner-Nordstrom spacetime. The circumferential radius of innermost stable circular orbit and the corresponding angular momentum of the test particles increase for α≠ 0.     

Geodesics of Random Riemannian Metrics

We analyze the disordered Riemannian geometry resulting from random perturbations of the Euclidean metric. We focus on geodesics, the paths traced out by a particle traveling in this quenched random environment. By taking the point of the view of the particle, we show that the law of its observed environment is absolutely continuous with respect to the law of the random metric, and we provide an explicit form for its Radon–Nikodym derivative. We use this result to prove a “local Markov property” along an unbounded geodesic, demonstrating that it eventually encounters any type of geometric phenomenon. We also develop in this paper some general results on conditional Gaussian measures. Our Main Theorem states that a geodesic chosen with random initial conditions (chosen independently of the metric) is almost surely not minimizing. To demonstrate this, we show that a minimizing geodesic is guaranteed to eventually pass over a certain “bump surface,” which locally has constant positive curvature. By using Jacobi fields, we show that this is sufficient to destabilize the minimizing property.     

Conformal Geodesics on Vacuum Space-times

 A classification of discrete integrable systems on quad–graphs, i.e. on surface cell decompositions with quadrilateral faces, is given. The notion of integrability laid in the basis of the classification is the three–dimensional consistency. This property yields, among other features, the existence of the discrete zero curvature representation with a spectral parameter. For all integrable systems of the obtained exhaustive list, the so called three–leg forms are found. This establishes Lagrangian and symplectic structures for these systems, and the connection to discrete systems of the Toda type on arbitrary graphs. Generalizations of these ideas to the three–dimensional integrable systems and to the quantum context are also discussed. Received: 14 February 2002 / Accepted: 22 September 2002 Published online: 8 January 2003 Acknowledgements. This research was partly supported by DFG (Deutsche Forschungsgemeinschaft) in the frame of SFB 288 ``Differential Geometry and Quantum Physics'. V.A. was also supported by the RFBR grant 02-01-00144. He thanks TU Berlin for warm hospitality during the visit when part of this work has been fulfilled. Communicated by L. Takhtajan     

Null Geodesics in Brane World Scenarios

We study the null bulk geodesic motion in the brane world in which the bulk metric has an un-stabilized extra spatial dimension. We find that the null bulk geodesic motion as observed on the 3-brane with Z₂ symmetry would be a timelike geodesic motion even though there exists an extra non-gravitational force in contrast with the case of the stabilized extra spatial dimension. In other words the presence of the extra non-gravitational force would not violate the Z₂ symmetry.     

Geodesics on the ellipsoid and monodromy

After reviewing the properties of the geodesic flow on the three-dimensional ellipsoid with distinct semi-axes, we investigate the three-dimensional ellipsoid with the two middle semi-axes equal, corresponding to a Hamiltonian invariant under rotations. The system is Liouville integrable, and symmetry reduction leads to a (singular) system on a two-dimensional ellipsoid with an additional potential and with a hard billiard wall inserted in the middle coordinate plane. We show that the regular part of the image of the energy–momentum map is not simply connected and there is an isolated critical value for zero angular momentum. The singular fibre of the isolated singular value is a doubly pinched torus multiplied by a circle. This circle is not a group orbit of the symmetry group, and thus analysis of this fibre is non-trivial. Finally we show that the system has a non-trivial monodromy, and consequently does not admit single-valued globally smooth action variables.