A thesis submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the degree of Doctor of Philosophy. July 2014. / This is the study of Lie algebraic properties of first integrals of scalar second-, third and
higher-order ordinary differential equations (ODEs). The Lie algebraic classification of such differential equations is now well-known from the works of Lie [10] as
well as recently Mahomed and Leach [19]. However, the algebraic properties of first
integrals are not known except in the maximal cases for the basic first integrals and
some of their quotients. Here our intention is to investigate the complete problem for
scalar second-order and maximal symmetry classes of higher-order ODEs using Lie
algebras and Lie symmetry methods. We invoke the realizations of low-dimensional
Lie algebras.
Symmetries of the fundamental first integrals for scalar second-order ODEs which are
linear or linearizable by point transformations have already been obtained. Firstly we
show how one can determine the relationship between the point symmetries and the
first integrals of linear or linearizable scalar ODEs of order two. Secondly, a complete
classi cation of point symmetries of first integrals of such linear ODEs is studied. As a
consequence, we provide a counting theorem for the point symmetries of first integrals
of scalar linearizable second-order ODEs. We show that there exists the 0, 1, 2 or 3
point symmetry cases. It is proved that the maximal algebra case is unique.
By use of Lie symmetry group methods we further analyze the relationship between the
first integrals of the simplest linear third-order ODEs and their point symmetries. It
is well-known that there are three classes of linear third-order ODEs for maximal and
submaximal cases of point symmetries which are 4, 5 and 7. The simplest scalar linear
third-order equation has seven point symmetries. We obtain the classifying relation
between the symmetry and the first integral for the simplest equation. It is shown
that the maximal Lie algebra of a first integral for the simplest equation y000 = 0 is
unique and four-dimensional. Moreover, we show that the Lie algebra of the simplest
linear third-order equation is generated by the symmetries of the two basic integrals.
We also obtain counting theorems of the symmetry properties of the first integrals for
such linear third-order ODEs of maximal type. Furthermore, we provide insights into
the manner in which one can generate the full Lie algebra of higher-order ODEs of
maximal symmetry from two of their basic integrals.
The relationship between rst integrals of sub-maximal linearizable third-order ODEs
and their symmetries are investigated as well. All scalar linearizable third-order equations
can be reduced to three classes by point transformations. We obtain the
classifying relations between the symmetries and the first integral for sub-maximal
cases of linear third-order ODEs. It is known, from the above, that the maximum Lie
algebra of the first integral is achieved for the simplest equation. We show that for
the other two classes they are not unique. We also obtain counting theorems of the
symmetry properties of the rst integrals for these classes of linear third-order ODEs.
For the 5 symmetry class of linear third-order ODEs, the first integrals can have 0,
1, 2 and 3 symmetries and for the 4 symmetry class of linear third-order ODEs they
are 0, 1 and 2 symmetries respectively. In the case of sub-maximal linear higher-order
ODEs, we show that their full Lie algebras can be generated by the subalgebras of
certain basic integrals. For the n+2 symmetry class, the symmetries of the rst integral
I2 and a two-dimensional subalgebra of I1 generate the symmetry algebra and for
the n + 1 symmetry class, the full algebra is generated by the symmetries of I1 and a
two-dimensional subalgebra of the quotient I3=I2.
Finally, we completely classify the first integrals of scalar nonlinear second-order ODEs
in terms of their Lie point symmetries. This is performed by first obtaining the classifying
relations between point symmetries and first integrals of scalar nonlinear second order
equations which admit 1, 2 and 3 point symmetries. We show that the maximum
number of symmetries admitted by any first integral of a scalar second-order nonlinear
(which is not linearizable by point transformation) ODE is one which in turn provides
reduction to quadratures of the underlying dynamical equation. We provide physical
examples of the generalized Emden-Fowler, Lane-Emden and modi ed Emden equations.
| Identifer | oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:wits/oai:wiredspace.wits.ac.za:10539/16838 |
| Date | 02 February 2015 |
| Creators | Mahomed, Komal Shahzadi |
| Source Sets | South African National ETD Portal |
| Language | English |
| Detected Language | English |
| Type | Thesis |
| Format | application/pdf |
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