Delay Equations and Radiation Damping

Starting from delay equations that model field retardation effects, we study the origin of runaway modes that appear in the solutions of the classical equations of motion involving the radiation reaction force. When retardation effects are small, we argue that the physically significant solutions belong to the so-called slow manifold of the system and we identify this invariant manifold with the attractor in the state space of the delay equation. We demonstrate via an example that when retardation effects are no longer small, the motion could exhibit bifurcation phenomena that are not contained in the local equations of motion.Comment: 15 pages, 1 figure, a paragraph added on page 5; 3 references adde

Lorentz Covariant Theory of Light Propagation in Gravitational Fields of Arbitrary-Moving Bodies

The Lorentz covariant theory of propagation of light in the (weak) gravitational fields of N-body systems consisting of arbitrarily moving point-like bodies with constant masses is constructed. The theory is based on the Lienard-Wiechert presentation of the metric tensor. A new approach for integrating the equations of motion of light particles depending on the retarded time argument is applied. In an approximation which is linear with respect to the universal gravitational constant, G, the equations of light propagation are integrated by quadratures and, moreover, an expression for the tangent vector to the perturbed trajectory of light ray is found in terms of instanteneous functions of the retarded time. General expressions for the relativistic time delay, the angle of light deflection, and gravitational red shift are derived. They generalize previously known results for the case of static or uniformly moving bodies. The most important applications of the theory are given. They include a discussion of the velocity dependent terms in the gravitational lens equation, the Shapiro time delay in binary pulsars, and a precise theoretical formulation of the general relativistic algorithm of data processing of radio and optical astrometric measurements in the non-stationary gravitational field of the solar system. Finally, proposals for future theoretical work being important for astrophysical applications are formulated.Comment: 77 pages, 7 figures, list of references is updated, to be published in Phys. Rev. D6

An application of Green-function methods to gravitational radiation theory

Previous work in the literature has studied gravitational radiation in black-hole collisions at the speed of light. In particular, it had been proved that the perturbative field equations may all be reduced to equations in only two independent variables, by virtue of a conformal symmetry at each order in perturbation theory. The Green function for the perturbative field equations is here analyzed by studying the corresponding second-order hyperbolic operator with variable coefficients, instead of using the reduction method from the retarded flat-space Green function in four dimensions. After reduction to canonical form of this hyperbolic operator, the integral representation of the solution in terms of the Riemann function is obtained. The Riemann function solves a characteristic initial-value problem for which analytic formulae leading to the numerical solution are derived.Comment: 18 pages, Revtex4. Submitted to Lecture Notes of S.I.M., volume edited by D. Cocolicchio and S. Dragomir, with kind permission by IOP to use material in Ref. [12]. arXiv admin note: substantial text overlap with arXiv:gr-qc/010107

A New Way to Make Waves

I describe a new algorithm for solving nonlinear wave equations. In this approach, evolution takes place on characteristic hypersurfaces. The algorithm is directly applicable to electromagnetic, Yang-Mills and gravitational fields and other systems described by second differential order hyperbolic equations. The basic ideas should also be applicable to hydrodynamics. It is an especially accurate and efficient way for simulating waves in regions where the characteristics are well behaved. A prime application of the algorithm is to Cauchy-characteristic matching, in which this new approach is matched to a standard Cauchy evolution to obtain a global solution. In a model problem of a nonlinear wave, this proves to be more accurate and efficient than any other present method of assigning Cauchy outer boundary conditions. The approach was developed to compute the gravitational wave signal produced by collisions of two black holes. An application to colliding black holes is presented.Comment: In Proceeding of CIMENICS 2000, The Vth International Congress on Numerical Methods in Engineering and Applied Science (Puerto La Cruz, Venezuela, March 2000

Factorization and reduction methods for optimal control of distributed parameter systems

A Chandrasekhar-type factorization method is applied to the linear-quadratic optimal control problem for distributed parameter systems. An aeroelastic control problem is used as a model example to demonstrate that if computationally efficient algorithms, such as those of Chandrasekhar-type, are combined with the special structure often available to a particular problem, then an abstract approximation theory developed for distributed parameter control theory becomes a viable method of solution. A numerical scheme based on averaging approximations is applied to hereditary control problems. Numerical examples are given

Legendre-Tau approximations for functional differential equations

The numerical approximation of solutions to linear functional differential equations are considered using the so called Legendre tau method. The functional differential equation is first reformulated as a partial differential equation with a nonlocal boundary condition involving time differentiation. The approximate solution is then represented as a truncated Legendre series with time varying coefficients which satisfy a certain system of ordinary differential equations. The method is very easy to code and yields very accurate approximations. Convergence is established, various numerical examples are presented, and comparison between the latter and cubic spline approximations is made