Some Aspects of Symmetries of Differential Equations and their Connection with the Underlying Geometry

Abstract

In this thesis symmetry methods have been used to solve some differential equations and to find the connection of isometries of some spaces with the symmetries of some related differential (geodesic) equations. It is proved here that the McVittie solution and its non-static analogue are the only plane symmetric spacetimes with electromagnetic field. The Einstein equations for nonstatic, shear-free, spherically symmetric, perfect fluid distributions reduce to one secondorder non-linear differential equation in the radial parameter. General solution of this equation is obtained in [11] by symmetry analysis. Corrections of some examples of the solution in the earlier work [11], by formulating a general requirement for physical relevance of the solution, are presented. An algebraic proof that the Lie algebra of generators of the system of n differential equations, (yO l' = 0, is isomorphic to the Lie algebra of the special linear group of order (n + 2), over the real numbers, is provided. A connection between the symmetries of manifolds and their geodesic equations, which are systems of second order ordinary differential equations, is sought through the geodesic equations of maximally symmetric spaces. Since such spaces have either constant positive, constant negative or zero curvature, three cases are considered. It is proved that for a space admitting so{n + 1) or so{n,l) as the maximal isometry algebra, the symmetry of the geodesic equations of the space is given by so{n + 1)E9 d2 or so{n, 1)E9 d2 (where d2 is the 2-dimensional dilation algebra), while for those admitting so{n) E9 s /R11 the algebra is sl{n + 2). A corresponding result holds on replacing so{n) by so{p, q) with p + q = n. It is conjectured that if the isometry algebra of any underlying space of non-zero curvature is h, then the Lie symmetry algebra of the geodesic equations is given by h EB d 2 provided that there is no cross-section of zero curvature. Some results on the Lie symmetry generators of equations with a small parameter and the relationship between symmetries and conservation laws for such equations are used to construct first integrals and Lagrangians for autonomous weakly non-linear systems. An adaptation of a theorem that provides the generators that leave the functional involving a Lagrangian for such equations is presented. A detailed example to illustrate the method is given.