Helmholtz conditions,covariance, and invariance identities

This paper is concerned with the problem of finding a multiplier matrixg which converts a prescribed system of second-order ordinary differential equations to the Euler-Lagrange form. Sufficient conditions for the existence of a multiplier matrix are given in the form of an infinite system of linear algebraic equations, provided the entries ofg may be regarded as components of a (0, 2) symmetric tensor field. As an application, conditions for the local existence of a metric tensor compatible with a given torsion-free connection are deduced.     

Fermion Fields,Boson Fields,and Gravitational Field

If a tetrad theory is derivable from a variational principle with a Lagrangian ?? of the form ?? = ??_F+??_M 6 tetrad components will be defined by the vacuum equations if the energy momentum tensor is symmetric. Therefore, we look for a realisation of a programme proposed in a little different way by TREDER according to which the 16 tetrad field equations should degenerate to 10 equations for the Riemannian metric if boson fields are the only source of the gravitational field.     

Poisson Involutions, Spin Calogero-Moser Systems Associated with Symmetric Lie Subalgebras and the Symmetric Space Spin Ruijsenaars-Schneider Models

We develop a general scheme to construct integrable systems starting from realizations in symmetric coboundary dynamical Lie algebroids and symmetric coboundary dynamical Poisson groupoids. The method is based on the successive use of Dirac reduction and Poisson reduction. Then we show that certain spin Calogero-Moser systems associated with symmetric Lie subalgebras can be studied in this fashion. We also consider some spin-generalized Ruisjenaars-Schneider equations which correspond to the N-soliton solutions of affine Toda field theory. In this case, we show how the equations are obtained from the Dirac reduction of some Hamiltonian system on a symmetric coboundary dynamical Poisson groupoid.     

On Spherically Symmetric Solutions¶of the Compressible Isentropic Navier–Stokes Equations

We prove the global existence of weak solutions to the Cauchy problem for the compressible isentropic Navier–Stokes equations in ℝ ⁿ (n= 2, 3) when the Cauchy data are spherically symmetric. The proof is based on the exploitation of the one-dimensional feature of symmetric solutions and use of a new (multidimensional) property induced by the viscous flux. The present paper extends Lions' existence theorem [15] to the case 1< γ <γ _n for spherically symmetric initial data, where γ is the specific heat ratio in the pressure, γ _n = 3/2 for n= 2 and γ _n = 9/5 for n= 3. Dedicated to Professor Rolf Leis on the occasion of his 70th birthday Received: 17 January 2000 / Accepted: 3 July 2000     

Equations of state for metals (Al, Fe, Cu, Pb), polyethylene, carbon, and boron nitride as applied to problems of dynamical compression

Metals (Al, Fe, Cu, Pb), polyethylene, and other plastic materials with a density of about 1 g/cm³ are commonly used as liners and screens in solving dynamic-compression problems that involve phase transitions. In this paper, the equations of state are presented in the form of formulas, graphs, and tables for the pressurep and energyE as functions of temperatureT and density ρ. These equations have a meaningful theoretical form and are based on the measured initial sound velocityc ₀, densityρ ₀, Gruneisen parameter Γ, heat capacityc _p, sublimation energyU _evp, and the known pressure dependence of the compression modulus ϱK/ϱp. These equations of state are in satisfactory agreement with available experimental data on shock compression. According to the same scheme, the equations of state are derived for carbon and boron nitride. However, in this case, the situation turned out to be much more complicated due to the existence of phase transitions from the hexagonal form into wurtzite and cubic forms. In deriving the equation of state, the equilibrium curve between the graphite-like and diamond phases on the phase diagram was additionally used. As a result of realization of the aforementioned scheme, the equations of state obtained (i.e., formulas, graphs, and tables) are in satisfactory agreement with experimental data. Translated from Preprint of the P. N. Lebedev Physical Institute No. 28 (1996).     

Computing multi-valued physical observables for the high frequency limit of symmetric hyperbolic systems

We develop a level set method for the computation of multi-valued physical observables (density, velocity, energy, etc.) for the high frequency limit of symmetric hyperbolic systems in any number of space dimensions. We take two approaches to derive the method.The first one starts with a weakly coupled system of an eikonal equation for phase S and a transport equation for density ρ:

The main idea is to evolve the density near the n-dimensional bi-characteristic manifold of the eikonal (Hamiltonian–Jacobi) equation, which is identified as the common zeros of n level set functions in phase space . These level set functions are generated from solving the Liouville equation with initial data chosen to embed the phase gradient. Simultaneously, we track a new quantity f = ρ(t,x,k)|det(_k)| by solving again the Liouville equation near the obtained zero level set = 0 but with initial density as initial data. The multi-valued density and higher moments are thus resolved by integrating f along the bi-characteristic manifold in the phase directions.The second one uses the high frequency limit of symmetric hyperbolic systems derived by the Wigner transform. This gives rise to Liouville equations in the phase space with measure-valued solution in its initial data. Due to the linearity of the Liouville equation we can decompose the density distribution into products of function, each of which solves the Liouville equation with L^∞ initial data on any bounded domain. It yields higher order moments such as energy and energy flux.The main advantages of these new approaches, in contrast to the standard kinetic equation approach using the Liouville equation with a Dirac measure initial data, include: (1) the Liouville equations are solved with L^∞ initial data, and a singular integral involving the Dirac-δ function is evaluated only in the post-processing step, thus avoiding oscillations and excessive numerical smearing; (2) a local level set method can be utilized to significantly reduce the computation in the phase space. These methods can be used to compute all physical observables for multi-dimensional problems.Our method applies to the wave fields corresponding to simple eigenvalues of the dispersion matrix. One such example is the wave equation, which will be studied numerically in this paper.     

Magnetisierungsprozesse in Kollektiven von Einbereichsteilchen. II

Certain mathematical aspects of the static pion-nucleon theory are investigated. We start from the fact that the theory in its uniformized form (cuts transformed away) leads to a system of functional equations for the S-matrix. The nonlinear mapping involved in the functional equations is a second-order Cremona transformation. After a summary of the general properties of Cremona transformations, the special transformations are studied which arise in the symmetric scalar and the Chew-Low theory respectively. The emphasis is on the possibility to separate, by means of a finite-order Cremona transformation, the functional equations into a set of uncoupled ones. For the symmetric scalar theory, the separation is trivial. For the Chew-Low theory, a proof of nonseparability is given.     

On the occurrence of naked singularity in spherically symmetric gravitational collapse

Generalizing earlier results of [1], we analyze here the spherically symmetric gravitational collapse of a matter cloud with a general form of matter for the formation of a naked singularity. It is shown that this is related basically to the choice of initial data to the Einstein field equations, and would therefore occur in generic situations from regular initial data within the general context considered here, subject to the matter satisfying the weak energy condition. The condition on initial data which leads to the formation of black hole is also characterized.     

A minimal energy tracking method for non-radially symmetric solutions of coupled nonlinear Schrödinger equations

We aim at developing methods to track minimal energy solutions of time-independent m-component coupled discrete nonlinear Schrödinger (DNLS) equations. We first propose a method to find energy minimizers of the 1-component DNLS equation and use it as the initial point of the m-component DNLS equations in a continuation scheme. We then show that the change of local optimality occurs only at the bifurcation points. The fact leads to a minimal energy tracking method that guides the choice of bifurcation branch corresponding to the minimal energy solution curve. By combining all these techniques with a parameter-switching scheme, we successfully compute a non-radially symmetric energy minimizer that can not be computed by existing numerical schemes straightforwardly.     

Non-Existence of Black Hole Solutions¶for a Spherically Symmetric, Static Einstein–Dirac–Maxwell System

We consider for j=?, … a spherically symmetric, static system of (2j+1) Dirac particles, each having total angular momentum j. The Dirac particles interact via a classical gravitational and electromagnetic field. The Einstein–Dirac–Maxwell equations for this system are derived. It is shown that, under weak regularity conditions on the form of the horizon, the only black hole solutions of the EDM equations are the Reissner–Nordstr?m solutions. In other words, the spinors must vanish identically. Applied to the gravitational collapse of a “cloud” of spin-?-particles to a black hole, our result indicates that the Dirac particles must eventually disappear inside the event horizon. Received: 2 November 1998 / Accepted: 23 February 1999     

Dissipation, decoherence and preparation effects in the spin-boson system

The dynamics of the reduced density matrix of the driven dissipative two-state system is studied for a general diagonal/off-diagonal initial state. We derive exact formal series expressions for the populations and coherences and show that they can be cast into the form of coupled nonconvolutive exact master equations and integral relations. We show that neither the asymptotic distributions, nor the transition temperature between coherent and incoherent motion, nor the dephasing rate and relaxation rate towards the equilibrium state depend on the particular initial state chosen. However, in the underdamped regime, effects of the particular initial preparation, e.g. in an off-diagonal state of the density matrix, strongly affect the transient dynamics. We find that an appropriately tuned external ac-field can slow down decoherence and thus allow preparation effects to persist for longer times than in the absence of driving. Received 23 October 1998 and Received in final form 26 February 1999     

Marginally Trapped Tubes Generated from Nonlinear Scalar Field Initial Data

We show that the maximal future development of asymptotically flat spherically symmetric black hole initial data for a self-gravitating nonlinear scalar field, also called a Higgs field, contains a connected, achronal, spherically symmetric marginally trapped tube which is asymptotic to the event horizon of the black hole, provided the initial data is sufficiently small and decays like O(r^-\frac12){O(r^{-\frac{1}{2}})}, and the potential function V is nonnegative with bounded second derivative. This result can be loosely interpreted as a statement about the stability of ‘nice’ asymptotic behavior of marginally trapped tubes under certain small perturbations of Schwarzschild.     

Eigenvalue-Dynamics off the Calogero–Moser System

By finding N(N– 1)/2 suitable conserved quantities, free motions of real symmetric N×N matrices X(t), with arbitrary initial conditions, are reduced to nonlinear equations involving only the eigenvalues of X – in contrast to the rational Calogero-Moser system, for which [X(0),Xd(0)] has to be purely imaginary, of rank one.     

Spherically symmetric vacuum solutions of modified gravity theory in higher dimensions

In this paper we investigate spherically symmetric vacuum solutions of f(R) gravity in a higher-dimensional spacetime. With this objective we construct a system of non-linear differential equations whose solutions depend on the explicit form assumed for the function F(R)=\fracdf(R)dRF(R)=\frac{df(R)}{dR} . We explicit show that for specific classes of this function exact solutions from the field equations are obtained; also we find approximated results for the metric tensor for more general cases admitting F(R) close to the unity.     

Detection of computer generated gravitational waves in numerical cosmologies

We propose to study the behavior of complicated numerical solutions to Einstein's equations for generic cosmologies by following the geodesic motion of a swarm of test particles. As an example, we consider a cylinder of test particles initially at rest in the plane symmetric Gowdy universe onT ³×R. For a circle of test particles in the symmetry plane, the geodesic equations predict evolution of the circle into distortions and rotations of an ellipse as well as motion perpendicular to the plane. The evolutionary sequence of ellipses depends on the initial position of the circle of particles. We display snapshots of the evolution of the cylinder.     

Dynamical Collapse of White Dwarfs in Hartree- and Hartree-Fock Theory

We study finite-time blow-up for pseudo-relativistic Hartree- and Hartree-Fock equations, which are model equations for the dynamical evolution of white dwarfs. In particular, we prove that radially symmetric initial configurations with negative energy lead to finite-time blow-up of solutions. Furthermore, we derive a mass concentration estimate for radial blow-up solutions. Both results are mathematically rigorous and are in accordance with Chandrasekhar’s physical theory of white dwarfs, stating that stellar configurations beyond a certain limiting mass lead to “gravitational collapse” of these objects. Apart from studying blow-up, we also prove local well-posedness of the initial-value problem for the Hartree- and Hartree-Fock equations underlying our analysis, as well as global-in-time existence of solutions with sufficiently small initial data, corresponding to white dwarfs whose stellar mass is below the Chandrasekhar limit.     

Global spherically symmetric solutions to the equations of a viscous polytropic ideal gas in an exterior domain

We consider the equations of a viscous polytropic ideal gas in the domain exterior to a ball in ⁿ (n=2 or 3) and prove the global existence of spherically symmetric smooth solutions for (large) initial data with spherical symmetry. The large-time behavior of the solutions is also discussed. To prove the existence we first study an approximate problem in a bounded annular domain and then obtain a priori estimates independent of the boundedness of the annular domain. Letting the diameter of the annular domain tend to infinity, we get a global spherically symmetric solution as the limit.Dedicated to Professor Rolf Leis on the occasion of his 65th birthdaySupported by the SFB 256 of the Deutsche Forschungsgemeinschaft at the University of Boon.     

Interfacial reaction kinetics

We study irreversible A-B reaction kinetics at a fixed interface separating two immiscible bulk phases, A and B. Coupled equations are derived for the hierarchy of many-body correlation functions. Postulating physically motivated bounds, closed equations result without the need for ad hoc decoupling approximations. We consider general dynamical exponent z, where is the rms diffusion distance after time t. At short times the number of reactions per unit area, , is 2nd order in the far-field reactant densities . For spatial dimensions dabove a critical value , simple mean field (MF) kinetics pertain, where Q_b is the local reactivity. For low dimensions , this MF regime is followed by 2nd order diffusion controlled (DC) kinetics, , provided . Logarithmic corrections arise in marginal cases. At long times, a cross-over to 1st order DC kinetics occurs: . A density depletion hole grows on the more dilute A side. In the symmetric case (), when the long time decay of the interfacial reactant density, , is determined by fluctuations in the initial reactant distribution, giving . Correspondingly, A-rich and B-rich regions develop at the interface analogously to the segregation effects established by other authors for the bulk reaction . For fluctuations are unimportant: local mean field theory applies at the interface (joint density distribution approximating the product of A and B densities) and . We apply our results to simple molecules (Fickian diffusion, z=2) and to several models of short-time polymer diffusion (z>2). Received 8 June 1998 and Received in final form 10 September 1999