This thesis is submitted to the School of Mathematical and Statistical Sciences, Faculty of Natural Sciences, University of Natal, Durban, in fulfillment of the requirements for the degree of Master of Science. The group theoretic approach devised by Lie is one of the most successful techniques available for solving these equations. His knowledge of the word processing package, J}TEX, not to mention multiple proof readings of this manuscript, enabled me to submit this dissertation with pride.
Towards the turn of the seventeenth century, modern calculus developed, fueled by an era of outstanding scientists whose passion and dedication to an abstract model of the physical world led them to study differential equations. 34; is determined by the available symmetries of the equation (which produced the appropriate transformations). Thus, it was often possible to reduce these equations to algebraic expressions only with the knowledge of the Lie group, which should be inherited by the reduction of the higher-order equation [3].
Although the terms Lie Groups and Lie algebras retained their respectability within the scientific world, the solvability of differential equations would now be determined by the methods of a new and growing group of computer scientists. However, most of the work related to Lie's group theory was not researched and published by him.
Uses of symmetries
Outline
Lie Theory of Differential Equations
Definitions
For each parameter c in 5 the transformations are one-to-one on D. 5 with the law of composition 4J forms a group. LIE GROUP OF TRANSFORMATIONS: One-parameter Lie group of transformations is a group of transformations which, in addition to the above, satisfy the following:. SUBGROUP: A subgroup of G is a group formed by a subset of elements of G with the same law of composition.
LINEAR GROUPS: The complex general linear group GL(n, C) and the real general linear group GL(n, R) consist of all nonsingular complex and real n x n matrices, respectively [29]. The latter can be considered a subgroup of the former.) The complex special linear group SL(n, C) is the subgroup of GL(n, C) consisting of matrices with determinant one. ROTATION GROUP: The rotation group SO(n,R) is the special or proper real orthogonal group given by the intersection of the group of orthogonal matrices O(n, R) and the complex special linear group, i.e. Note: Any Abelian algebra is solvable, and any Lie algebra of dimension :S 3 is solvable, except when dimL = 3= dimL'.
While we can define a Lie algebra over any field, in practice it is usually considered over real and complex fields. Lie's main result [33] is the proof that it is always possible to assign a continuous group (Lie group) to a corresponding Lie algebra and vice versa.
The Algorithm
The Jacobi identity plays the same role for Lie algebras that the associative law plays for associative algebras. In the case of point symmetry we require that the coefficient functions ~ and TJ depend only on the independent and dependent variables (in this case x and y). Thus, we are investigating a fourth-order ordinary differential equation invariant only under two-point symmetry.
Usually the number of symmetries present (which is all we are concerned with if the Lie algebra is solvable) implies that the equation cannot be reduced to quadratures. 7e actually considers C~lJ, the first extension of C2. Da-.1: is not a variable relevant to the reduced equation, we ignore. Since reduction via X2 results in loss of Xl as a point symmetry of the reduced equation, we attempt reduction via Xl, in which case.
Reduction via Ut results in a first-order ordinary differential equation with (at least) one-point symmetry. Reversing the transformations and (2.12) and integrating the resulting first-order ordinary differential equations will give the solution of the original equation.
Summary
Reduction of Third Order Equations
Introduction
We begin our analysis by considering ordinary differential equations of the third order, invariant with respect to two- and three-dimensional Lie algebras of point symmetries. We reduce each differential equation of the third order through each of its symmetries and consider the symmetries of the resulting differential equations of the second order.
Two-Dimensional Lie AIgebras
The remammg symmetry, C2, can be rewritten in terms of the new coordinates. and is a symmetry of the newly formed second order ordinary differential equation, ie 3.12) Equation (3.11) has no further symmetries and can therefore only be reduced to a first-order differential equation. The invariants of the symmetry (3.7) are. and we use them as the new variables. Here and in what follows, the variables Ui and Vi refer to the reduction variables obtained from the symmetry Cd. 3.41) Equation (3.40) has no further symmetries and can therefore only be reduced to a first-order differential equation. The results obtained in this section show that the third-order equation invariant only under the two-dimensional Lie algebras cannot simply be reduced to second-order equations with more than one symmetry.
Three-dimensional Lie algebras
- A 3,8 III
Thus, one can now see to what extent the solvability of a differential equation depends on the order of the symmetries used for reduction. Therefore, if one obtains a second-order equation of the form (3.75) or (3.80), one can increase its order by one, (thus achieving asymmetry), and then reduce the third-order differential equation via Cl. Since Cl as well as C2 cannot be written in terms of the new coordinates, u and V, the second-order differential equation (3.104) will lose both remaining symmetries.
In both cases, the new second-order equations contain two symmetries and can be reduced to quadratures. Thus, none of the remaining symmetries will be lost via this reduction route. which reduces the original equation to. Since neither Cl nor C2 transforms to symmetries in the new variables, the newly formed second order ordinary differential equation cannot be further reduced.
Thus, if one is presented with an equation of the form (3.149), one only needs to perform the above transformations to convert it into a form that can be reduced to quadrature. Transforming the second-order differential equation (3.196) into the form of (3.184) or (3.190) will increase the number of symmetries it contains to two. Following the already familiar path, we start by reducing equation (3.223) through the symmetry Gl.
None of the t\vo remaining symmetries, namely G l or G2, can be rewritten in terms of these new coordinates, so reduction via G3 cannot be an optimal method for solving the differential equation (3.223). To solve an equation of the form (3.242), one of the above transformations must be used. G3 cannot be rewritten according to the new coordinates; only G2 transforms into the symmetry of equation (3.370), i.e.
3.472) Further reduction is not possible due to the fact that none of the remaining symmetries can be rewritten in terms of the new variables. 3.479) Reduction to quadrature is therefore only possible if the second-order differential equations, (3.472) and (3.477), are transformed into an equation of the form (3.466) using the above transformations.
Conclusion
4.127) We submit this list of equations as a contribution to the class of second-order ordinary differential equations that can be reduced to quadrature. It remains to consider other third-order invariant equations among larger (>3) dimensional Lie groups. We conclude by saying that mathematics is indeed a vital and necessary step in the exploration of the physical world.
The accuracy of the results it produces plays a major role in reducing time, an uncontrollable variable, often considered man's worst enemy.
Bibliography
9] GonzaJez-Gascon F and GonzaJez-Lopez A, Newtonian systems of quadrature-integrable differential equations with trivial set of point symmetries Phys Lett A. 11] Govinder K 5 and Leach P G L, A group theoretic approach to a class of order second ordinary differential equations that do not possess lie point symmetry J Phase A: Math Gen.
