I= SOME ASPECTS OF SYMMETRIES OF DIFFERENTIAL EQUATIONS AND THEIR CONNECTION WITH THE UNDERLYING GEOMETRY
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Title of Thesis
SOME ASPECTS OF SYMMETRIES OF DIFFERENTIAL EQUATIONS AND THEIR CONNECTION WITH THE UNDERLYING GEOMETRY
Author(s)
Tooba Feroze
Institute/University/Department Details
Quaid-i-Azam University, Islamabad
Session
2004
Subject
Mathematics
Number of Pages
132
Keywords (Extracted from title, table of contents and abstract of thesis)
differential equations, underlying geometry, isometries, lie algebra, autonomous weakly non-linear systems, spacetimes, geodesic equations

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.
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5259.96 KB
S. No. Chapter Title of the Chapters Page Size (KB)
1 0 Contents
328.38 KB
2 1 Preliminaries 4
968.37 KB
  1.1 Introduction 4
  1.2 Manifolds and Vector Fields 12
  1.3 Lie Groups and Lie Algebras 14
  1.4 Lie Derivatives and Isometries 16
  1.5 Symmetries of Differential Equations 17
  1.6 Symmetries and Conservation Laws 22
3 2 Applications of Symmetry Methods in General Relativity 25
1085.48 KB
  2.1 Plane Symmetric Electromagnetic Solutions of the Einstein Field Equations 26
  2.2 Spacetimes Not Admitting Sourceless Electromagnetic Fields 29
  2.3 Spacetimes Admitting Sourceless Electromagnetic Fields 39
  2.4 Non-Static, Spheric ally Symmetric, Shear-Free, Perfect Fluid Solutions of Einstein's Equations 41
  2.5 The Physical Criteria 47
4 3 The Connection Between Isometries and Symmetries of Geodesic Equations of the Underlying Spaces 53
1034.5 KB
  3.1 The Lie algebra of a Second Order Vector Differential Equation ( ya ) = 0 54
  3.2 The Algebra of ( ya ) = 0 55
  3.3 Symmetries of the Geodesic Equations of Maximally Symmetric 2 Dimensional Spaces 57
  3.4 Symmetries of the Geodesic Equations of Less Symmetric 2- Dimensional Spaces 61
  3.5 Symmetries of the Geodesic Equations of Maximally Symmetric Higher Dimensional Spaces 64
  3.6 Symmetries of the Reduced System of the Geodesic Equations of Maximally Symmetric Higher Dimensional Spaces 80
5 4 Approximate Symmetries 83
533.16 KB
  4.1 Algorithm for Calculating Infinitesimal Approximate Symmetries 85
  4.2 Applications 88
  4.3 Approximate Symmetries of y" + eF (x) y' + y = f (y' ) 97
6 5 Conclusion 99
463.56 KB
  5.1 Uniqueness of the McVittie Solution 100
  5.2 Physics of Some New Solutions of the Einstein Field Equations 101
  5.3 Isometries of the Maximally Symmetric Spaces and Symmetries of their Geodesic Equations 102
  5.4 Some Discussion on Approximate Symmetries 106
7 6 References 108
123.44 KB
8 7 Work Already Published from the Thesis 111
875.63 KB