SOME ASPECTS OF SYMMETRIES OF DIFFERENTIAL EQUATIONS AND THEIR CONNECTION WITH THE UNDERLYING GEOMETRY

Tooba, Feroze (2004) SOME ASPECTS OF SYMMETRIES OF DIFFERENTIAL EQUATIONS AND THEIR CONNECTION WITH THE UNDERLYING GEOMETRY. Doctoral thesis, Quaid-i-Azam University, Islamabad.

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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 Mc Vittie solution and its non-static analogue are the only plane symmetric space times with electromagnetic field. The Einstein equations for non static, shear-free, spherically symmetric, perfect fluid distributions reduce to one second order 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) = 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)$ d2 or so(n, 1)$ d2 (where d2 is the 2-dimensional dilation algebra), while for those admitting so(n) $ J?II 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 El:) d2 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.

Item Type: Thesis (Doctoral)
Uncontrolled Keywords: differential equations, underlying geometry, isometries, lie algebra, autonomous weakly non-linear systems, spacetimes, geodesic equations
Subjects: Q Science > QA Mathematics
Depositing User: Muhammad Khan Khan
Date Deposited: 29 Sep 2016 10:02
Last Modified: 29 Sep 2016 10:02
URI: http://eprints.hec.gov.pk/id/eprint/1863

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