On the Lie symmetries of a class of generalized Ermakov systems

The symmetry analysis of Ermakov systems is extended to the generalized case where the frequency depends on the dynamical variables besides time. In this extended framework, a whole class of nonlinearly coupled oscillators are viewed as Hamiltonian Ermakov system and exactly solved in closed form

Lie Point Symmetries for Reduced Ermakov Systems

Reduced Ermakov systems are defined as Ermakov systems restricted to the level surfaces of the Ermakov invariant. The condition for Lie point symmetries for reduced Ermakov systems is solved yielding four infinite families of systems. It is shown that SL(2,R) always is a group of point symmetries for the reduced Ermakov systems. The theory is applied to a model example and to the equations of motion of an ion under a generalized Paul trap

Thermal Behaviour of Euclidean Stars

A recent study of dissipative collapse considered a contracting sphere in which the areal and proper radii are equal throughout its evolution. The interior spacetime was matched to the exterior Vaidya spacetime which generated a temporal evolution equation at the boundary of the collapsing sphere. We present a solution of the boundary condition which allows the study of the gravitational and thermodynamical behaviour of this particular radiating model.Comment: 10 pages, 3 figure

A group theoretic approach to shear-free radiating stars

A systematic analysis of the junction condition, relating the radial pressure with the heat flow in a shear-free relativistic radiating star, is undertaken. This is a highly nonlinear partial differential equation in general. We obtain the Lie point symmetries that leave the boundary condition invariant. Using a linear combination of the symmetries, we transform the junction condition into ordinary differential equations. We present several new exact solutions to the junction condition. In each case we can identify the exact solution with a Lie point generator. Some of the solutions obtained satisfy the linear barotropic equation of state. As a special case we regain conformally flat models which were found previously. Our analysis highlights the interplay between Lie algebras, nonlinear differential equations and application to relativistic astrophysics.Comment: 11 pages, Submitted for publication, minor revision