Radiating stars with generalised Vaidya atmospheres

We model the gravitational behaviour of a radiating star when the exterior geometry is the generalised Vaidya spacetime. The interior matter distribution is shear-free and undergoing radial heat flow. The exterior energy momentum tensor is a superposition of a null fluid and a string fluid. An analysis of the junction conditions at the stellar surface shows that the pressure at the boundary depends on the interior heat flux and the exterior string density. The results for a relativistic radiating star undergoing nonadiabatic collapse are obtained as a special case. For a particular model we demonstrate that the radiating fluid sphere collapses without the appearance of the horizon at the boundary.     

Lie symmetries of difference equations

The discrete heat equation is worked out to illustrate the search of symmetries of difference equations. Special attention it is paid to the Lie structure of these symmetries, as well as to their dependence on the derivative’s discretization. The case ofq-symmetries for discrete equations in aq-lattice is briefly considered at the end. Talk delivered by J. Negro at the DI-CRM Workshop held in Prague, 18–21 June 2000. This work has been partially supported by DGES of the Ministerio de Educación y Cultura of Spain under Projects PB98-0360 and the Junta de Castilla y León (Spain).     

离散Lagrange系统的Lie对称性

研究离散Lagrange系统的Lie对称性. 根据离散变分原理建立离散系统的运动方程. 给出离散运动方程Lie对称性的定义和确定方程. 举例说明结果的应用. 关键词： 离散Lagrange系统 离散变分原理 Lie对称性 确定方程     

Relativity symmetries and Lie algebra contractions

We revisit the notion of possible relativity or kinematic symmetries mutually connected through Lie algebra contractions under a new perspective on what constitutes a relativity symmetry. Contractions of an SO(m,n)$SO(m,n)$ symmetry as an isometry on an m+n$m+n$ dimensional geometric arena which generalizes the notion of spacetime are discussed systematically. One of the key results is five different contractions of a Galilean-type symmetry G(m,n)$G(m,n)$ preserving a symmetry of the same type at dimension m+n−1$m+n-1$, e.g.   a G(m,n−1)$G(m,n-1)$, together with the coset space representations that correspond to the usual physical picture. Most of the results are explicitly illustrated through the example of symmetries obtained from the contraction of SO(2,4)$SO(2,4)$, which is the particular case for our interest on the physics side as the proposed relativity symmetry for “quantum spacetime”. The contractions from G(1,3)$G(1,3)$ may be relevant to real physics.     

Lie Symmetries for a Conformally Flat Radiating Star

We consider a relativistic radiating spherical star in conformally flat spacetimes. In particular we study the junction condition relating the radial pressure to the heat flux at the boundary of the star which is a nonlinear partial differential equation. The Lie symmetry generators that leave the equation invariant are identified and we generate an optimal system. Each element of the optimal system is used to reduce the partial differential equation to an ordinary differential equation which is further analysed. We identify new categories of exact solutions to the boundary conditions. Two classes of solutions are of interest. The first class depends on a self similar variable. The second class is separable in the spacetime variables.     

Geodesic models generated by Lie symmetries

New theorems about the existence of solution for a system of infinite linear equations with a Vandermonde type matrix of coefficients are proved. Some examples and applications of these results are shown. In particular, a kind of these systems is solved and applied in the field of the General Relativity Theory of Gravitation. The solution of the system is used to construct a relevant physical representation of certain static and axisymmetric solution of the Einstein vacuum equations. In addition, a newtonian representation of these relativistic solutions is recovered. It is shown as well that there exists a relation between this application and the classical Haussdorff moment problem.     

Lie symmetries and superintegrability in quantum mechanics

Starting from the structure of the higher order Lie symmetries of the Schrödinger equation in the Euclidean plane E2, we establish, in the case of first-and second-order symmetries, the relations between separation of variables and superintegrable systems in quantum mechanics.     

Lie symmetries of quadratic two-dimensional difference equations

We find all systems of first-order quadratic autonomous two-dimensional difference equations which have two linear Lie symmetries. Knowledge of these symmetries permits the systems to be integrated by a reduction procedure.     

Lie symmetries and conserved quantities of Birkhoff systems with unilateral constraints

In this paper we study the Lie symmetries of Birkhoff systems with unilateral constraints.We give the conditions for,and the form of,conserved quantities due to the Lie symmetries of the systems.and we also study the inverse problem of the Lie symmetries of the systems.Finally,an example is given to illustrate the application of the results.     

Lie symmetries and solitons in nonlinear systems with spatially inhomogeneous nonlinearities

Using Lie group theory and canonical transformations, we construct explicit solutions of nonlinear Schr?dinger equations with spatially inhomogeneous nonlinearities. We present the general theory, use it to show that localized nonlinearities can support bound states with an arbitrary number solitons, and discuss other applications of interest to the field of nonlinear matter waves.     

The geometric nature of Lie and Noether symmetries

It is shown that the Lie and the Noether symmetries of the equations of motion of a dynamical system whose equations of motion in a Riemannian space are of the form [(x)\ddot]ⁱ+G_jkⁱ[(x)\dot]^j[(x)\dot] ^k+f(xⁱ)=0{\ddot{x}^{i}+\Gamma_{jk}^{i}\dot{x}^{j}\dot{x} ^{k}+f(x^{i})=0} where f(x ⁱ ) is an arbitrary function of its argument, are generated from the Lie algebra of special projective collineations and the homothetic algebra of the space respectively. Therefore the computation of Lie and Noether symmetries of a given dynamical system in these cases is reduced to the problem of computation of the special projective algebra of the space. It is noted that the Lie and Noether symmetry vectors are common to all dynamical systems moving in the same background space. The selection of the vectors which are Lie/Noether symmetries for a given dynamical system is done by means of a set of differential conditions involving the vectors and the potential function defining the dynamical system. The general results are applied to a number of different applications concerning (a) The motion in Euclidean space under the action of a general central potential (b) The motion in a space of constant curvature (c) The determination of the Lie and the Noether symmetries of class A Bianchi type hypersurface orthogonal spacetimes filled with a scalar field minimally coupled to gravity (d) The analytic computation of the Bianchi I metric when the scalar field has an exponential potential.     

Lie symmetries and invariants of constrained Hamiltonian systems

According to the theory of the invariance of ordinary differential equations under the infinitesimal transformations of group, the relations between Lie symmetries and invariants of the mechanical system with a singular Lagrangian are investigated in phase space. New dynamical equations of the system are given in canonical form and the determining equations of Lie symmetry transformations are derived. The proposition about the Lie symmetries and invariants are presented. An example is given to illustrate the application of the result in this paper.     

Lie symmetries and exact solutions for a short-wave model

In this paper, the Lie symmetry analysis and generalized symmetry method are performed for a short-wave model. The symmetries for this equation are given, and the phase portraits of the traveling wave systems are analyzed using the bifurcation theory of dynamical systems. The exact parametric representations of four types of traveling wave solutions are obtained.     

Lie and Noether symmetries of geodesic equations and collineations

The Lie symmetries of the geodesic equations in a Riemannian space are computed in terms of the special projective group and its degenerates (affine vectors, homothetic vector and Killing vectors) of the metric. The Noether symmetries of the same equations are given in terms of the homothetic and the Killing vectors of the metric. It is shown that the geodesic equations in a Riemannian space admit three linear first integrals and two quadratic first integrals. We apply the results in the case of Einstein spaces, the Schwarzschild spacetime and the Friedman Robertson Walker spacetime. In each case the Lie and the Noether symmetries are computed explicitly together with the corresponding linear and quadratic first integrals.     

Lie algebra symmetries and quantum phase transitions in nuclei

In this paper, an overview of some aspects of quantum phase transitions (QPT) in nuclei is given and they are: (i) QPT in interacting boson model (sdIBM), (ii) QPT in two-level models, (iii) critical point E(5) and X(5) symmetries, (iv) QPT in a simple solvable model with three-body forces. In addition, some open problems are also given.     

Loop Algebra of Lie symmetries for a short-wave equation

It is demonstrated that Lie point symmetries associated with a nonlinear equation for short waves in three dimensions generate an infinite-dimensional Lie algebra—a loop Algebra. Classification of the independent sets of the subalgebra is done through the adjoint action of the corresponding generators. Different forms of similarity solutions are discussed.     

On geometric approach to Lie symmetries of differential-difference equations

Based upon Cartan's geometric formulation of differential equations, Harrison and Estabrook proposed a geometric approach for the symmetries of differential equations. In this Letter, we extend Harrison and Estabrook's approach to analyze the symmetries of differential-difference equations. The discrete exterior differential technique is applied in our approach. The Lie symmetry of (2+1)-dimensional Toda equation is investigated by means of our approach.     

Lie symmetries and conserved quantities of non—holonomic mechanical systems with unilateral Vacco constraints

In this paper,we study the Lie symmetries and the conserved quantities of non-holonomic mechanical systems with unilateral Vacco constraints.we give the conditions and the form of conserved quantities due to the Lie symmetries of the systems,and present the inverse problem of the above proble,i.e.finding the corresponding Lie symmetry transformation according to a given integral of the system.Finally,we give an example to illustrate the application of the results.