On the dynamics of a rigid body carrying a material point

In this paper we consider a system consisting of an outer rigid body (a shell) and an inner body (a material point) which moves according to a given law along a curve rigidly attached to the body. The motion occurs in a uniform field of gravity over a fixed absolutely smooth horizontal plane. During motion the shell may collide with the plane. The coefficient of restitution for an impact is supposed to be arbitrary. We present a derivation of equations describing both the free motion of the system over the plane and the instances where collisions with the plane occur. Several special solutions to the equations of motion are found, and their stability is investigated in some cases. In the case of a dynamically symmetric body and a point moving along the symmetry axis according to an arbitrary law, a general solution to the equations of free motion of the body is found by quadratures. It generalizes the solution corresponding to the classical regular precession in Euler??s case. It is shown that the translational motion of the shell in the free flight regime exists in a general case if the material point moves relative to the body according to the law of areas.     

A boundary value problem for the spherically symmetric motion of a pressureless gas with a temperature‐dependent viscosity

We consider an initial‐boundary value problem for the equations of spherically symmetric motion of a pressureless gas with temperature‐dependent viscosity µ(θ) and conductivity κ(θ). We prove that this problem admits a unique weak solution, assuming Belov's functional relation between µ(θ) and κ(θ) and we give the behaviour of the solution for large times. Copyright © 2009 John Wiley & Sons, Ltd.     

(2+1)-Dimensional Gravity Coupled to a Dust Shell: Quantization in Terms of Global Phase Space Variables

Theoretical and Mathematical Physics - We perform a canonical analysis of a model in which gravity is coupled to a spherically symmetric dust shell in 2+1 space-time dimensions. The result is a...     

Exact Solutions of Compressible Flow Equations with Spherical Symmetry

In this paper, we construct spherically symmetric solutions of the equations of compressible flow, which are important in the theory of explosion waves in air, water, and other media. Following McVittie [ 1 ], we write a general solution form, in terms of velocity potential, as a product of a function of time and a function of a similarity variable. First, we find solutions to the equations of motion and continuity without reference to adiabatic or isentropic relation. These solutions are quite general and can be applied to nonadiabatic motions, such as the motions of interstellar gas clouds that lose energy by radiation. All the solutions found by McVittie [ 1 ] have linear velocity profile with respect to distance. We introduce a nonlinear form of the velocity function containing an arbitrary function of the similarity variable. Adiabatic conditions lead to a second-order ODE, which we discuss in some detail. We relate our work to the earlier investigations of Taylor [ 2 ], McVittie [ 1 ], and Keller [ 3 ].     

On the acceleration of shock waves and concentration of energy

By a series of simple examples related to exact solutions of problems in gas dynamics and magnetohydrodynamics, possible mechanisms of acceleration of shock waves and concentration of energy are elucidated. The acceleration of a shock wave is investigated in the problem of motion of a plane piston at a constant velocity in the case when the initial density of the medium drops in the presence of constant counterpressure. It is shown that in this situation a “blow-up” regime is induced by a shock wave going to infinity in finite time even for limited work of the piston. A simple spherically symmetric solution with a converging shock wave is constructed and shown to lead to the concentration of energy. A general method for solving one-dimensional non-self-similar problems related to matching the equilibrium state to a motion with homogeneous deformation on a shock wave is discussed; this method leads to a solution in quadratures.     

一类对偶平坦的球对称的芬斯勒度量

In this paper,we study spherically symmetric Finsler metrics.By analysing the solution of the spherically symmetric dually flat equation,we construct several new families of dually flat spherically symmetric Finsler metrics.     

Isotropic geodesics in the neighborhood of a singularity of the static, spherically symmetric solution

The motion of a massless particle is studied in the neighborhood of the singularity of the static, spherically symmetric solution in relativistic gravitation theory. It is shown that the existence of a nonzero graviton mass results in the appearance of some qualitatively new trajectories of motion. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 111, No. 2, pp. 312–320, May, 1997.     

Cross section for capture of a particle by a spherically symmetric body and different types of finite motion in the relativistic theory of gravitation

Conclusions It is clear that all properties of the metric (1) that can be formulated in the language of its invariants are identical when these properties are considered in general relativity and in the RTG. For example, the expressions for the cross section for capture of particles by a black hole in general relativity and a sufficiently compact body in the RTG are identical. Similarly, when we consider finite motion of particles in the RTG and in general relativity there are analogous sets of different types of motion of the particles (there is only the characteristic difference in the coordinate r characterized by the relation (6)).We note that circular orbits in the gravitational field of a spherically symmetric body were considered in the framework of the RTG in [3], and it was found that these orbits exist for r>2 and are Lyapunov stable for r>5. A relation characterizing the Thomas precession identical to the corresponding expression obtained in general relativity was also obtained in [3]. Thus, differences between general relativity and the RTG can appear only in properties that are not formulated in the language of the invariants of the metric (1). Therefore, if, for example, we consider the problem of the scattering of a particle by a spherically symmetric compact body in the framework of the RTG and general relativity, then we cannot find a difference between the theories of gravitation, since the expressions for the capture cross sections are the same.Institute of Theretical and Experimental Physics. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 85, No. 1, pp. 150–154, October, 1990.     

SOME STATIC RELATIVISTIC SOLUTIONS

Abstract

We analyse static spherically symmetric spacetimes in general relativity. For a particular choice of one of the gravitational potentials we show that the solution of the Einstein field equations is reduced to generating a solution of the hypergeometric differential equation. Some exact solutions are presented in terms of special functions and we demonstrate, with the assistance of MATHEMATICA, that these solutions may be reexpressed in terms of elementary functions. We regain a well known static model from our results.     

General spherically symmetric Finsler metrics with constant Ricci and flag curvature

In this paper, we investigate the flag curvature of a special class of Finsler metrics called general spherically symmetric Finsler metrics, which are defined by a Euclidean metric and two related 1-forms. We find equations to characterize the class of metrics with constant Ricci curvature (tensor) and constant flag curvature. Moreover, we study general spherically symmetric Finsler metrics with the vanishing non-Riemannian quantity χ-curvature. In particular, we construct some new projectively flat Finsler metrics of constant flag curvature.     

Landweber iterative method for an inverse source problem of time-fractional diffusion-wave equation on spherically symmetric domain

In this paper, an inverse source problem of time-fractional diffusion-wave equation on spherically symmetric domain is considered. In general, this problem is ill-posed. Landweber iterative method is used to solve this inverse source problem. The error estimates between the regularization solution and the exact solution are derived by an a-priori and an a-posteriori regularization parameters choice rules. The numerical examples are presented to verify the efficiency and accuracy of the proposed methods.     

Barenblatt solution and spherically symmetric solutions of the porous media equation

We consider the Cauchy problem of the porous media equation. We show that it is spherically symmetric solution has the same property as Barenblatt solution, with respct to some regularity property.     

Birkhoff''s theorem on manifolds

Brikhoff's theorem states that if the geometry of a given region of space-time is first spherically symmetric and secondly a solution to the Einstein empty space equations, that then that geometry is a piece of the Schwarzschild geometry. Here we show that Birkhoff's theorem holds on differentiable manifolds in general.     

一类非线性发展方程组的定解问题

将不可压缩的广义neo-Hookean材料组成的超弹性圆柱壳径向对称运动的数学模型归结为一类非线性发展方程组的初边值问题.利用材料的不可压缩条件和边界条件求得了描述圆柱壳内表面径向运动的二阶非线性常微分方程.给出了微分方程的周期解(即圆柱壳的内表面产生非线性周期振动)的存在条件,讨论了材料参数和结构参数对方程的周期解的影响,并给出了相应的数值模拟.     

Gravitational field of an expanding spherically symmetric shell in RTG. Linear velocity approximation

The appearance of nonstatic behavior of the gravitational fieldof an expanding, spherically symmetric shell is shown in RTG under the linear velocity approximation for a gravitational field of arbitrary magnitude.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 108, No. 1, pp. 79–83. July, 1996.     

Stability for steady states of Navier-Stokes-Poisson equations

In this paper, we study the stationary solution and nonlinear stability of Navier-Stokes-Poisson equations. Using variational method, we construct steady states of the N-S-P system as minimizers of a suitably defined energy functional, then show their dynamical stability against general, i.e. not necessarily spherically symmetric perturbation.     

More about wormholes in generalized Galileon theories

We consider a class of generalized Galileon theories within general relativity in space–times with more than two spatial dimensions. We show that these theories do not admit stable, static, spherically symmetric, asymptotically flat Lorentzian wormholes.     

First integrals for charged perfect fluid distributions

We study the evolution of shear-free spherically symmetric charged fluids in general relativity. We find a new parametric class of solutions to the Einstein-Maxwell system of field equations. Our charged results are a generalisation of earlier treatments for neutral relativistic fluids. We regain the first integrals found previously for uncharged matter as a special case. In addition an explicit first integral is found which is necessarily charged.     

On Blowup for Time-Dependent Generalized Hartree–Fock Equations

We prove finite-time blowup for spherically symmetric and negative energy solutions of Hartree–Fock and Hartree–Fock–Bogoliubov-type equations, which describe the evolution of attractive fermionic systems (e.g. white dwarfs). Our main results are twofold: first, we extend the recent blowup result of Hainzl and Schlein (Comm. Math. Phys. 287:705–714, 2009) to Hartree–Fock equations with infinite rank solutions and a general class of Newtonian type interactions. Second, we show the existence of finite-time blowup for spherically symmetric solutions of a Hartree–Fock–Bogoliubov model, where an angular momentum cutoff is introduced. We also explain the key difficulties encountered in the full Hartree–Fock–Bogoliubov theory.     

一类射影平坦的球对称的芬斯勒度量

本文研究了射影平坦芬斯勒度量的构造问题.通过分析射影平坦的球对称的芬斯勒度量的方程的解,构造了一类新的射影平坦的芬斯勒度量,并得到了射影平坦的球对称的芬斯勒度量的射影因子和旗曲率.