A thesis submitted to the faculty of Science, University of the Witwatersrand, in
fulfilment of the requirements for the degree of Doctor of Philosophy, / The inverse problem in the Calculus of Variations involves determining the Lagrangians,
if any, associated with a given (system of) differential equation(s). One
can classify Lagrangians according to the Lie algebra of symmetries of the Action
integral (the Noether algebra). We give a complete classification of first-order Lagrangians
defined on the line and produce results pertaining to the dimensionality
of the algebra of Noether symmetries and compare and contrast these with similar
results on the algebra of Lie symmetries of the corresponding Euler-Lagrange .equations.
It is proved that the maximum dimension of the Noether point symmetry
algebra of a particle Lagrangian. is five whereas it is known that the maximum dimension
Qf the Lie algebra of the corresponding scalar second-order Euler-Lagrange
equation is eight. Moreover, we show th'a.t a particle Lagrangian does not admit a
maximal four-dimensional Noether point symmeiry algebra and consequently a particle
Lagrangian admits the maximal r E {O, 1,2,3, 5}-dimensional Noether point
symmetry algebra,
It is well .known that an important means of analyzing differential equations lies in
the knowledge of the first integrals of the equation. We deliver an algorithm for
finding first integrals of partial differential equations and show how some of the
symmetry properties of the first integrals help to 'further' reduce the order of the
equations and sometimes completely solve the equations.
Finally, we discuss some open questions. These include the inverse problem and
classification of partial differential equations. ALo, there is the question of the
extension of the results to 'higher' dimensions. / Andrew Chakane 2018
| Identifer | oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:wits/oai:wiredspace.wits.ac.za:10539/26093 |
| Date | January 1994 |
| Creators | Kara, A. H. |
| Source Sets | South African National ETD Portal |
| Language | English |
| Detected Language | English |
| Type | Thesis |
| Format | application/pdf |
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