By Peter J. Olver

Symmetry tools have lengthy been well-known to be of significant significance for the research of the differential equations. This ebook offers a high-quality creation to these purposes of Lie teams to differential equations that have proved to be worthwhile in perform. The computational equipment are provided in order that graduate scholars and researchers can without difficulty learn how to use them. Following an exposition of the functions, the publication develops the underlying idea. a few of the themes are offered in a unique manner, with an emphasis on particular examples and computations. extra examples, in addition to new theoretical advancements, look within the routines on the finish of every bankruptcy.

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**Additional info for Applications of Lie Groups to Differential Equations**

**Example text**

The continuous nature of the observed properties of gases, for example the uniform and constant pressure exerted by a gas on the walls of the containing vessel, is explained (intuitively and mathematically) as the average effect of an unimaginably large number of discrete processes effected by individual molecules (in the case of pressure, collisions with the walls of the vessel). In order to describe some of the changes in the conceptual structure of physics brought about by the development of quantum theory, in terms of the concept of model, it is necessary to sketch the two particular models developed by classical physics, the particle model and the field model.

Disintegration of coherence of descriptions with changes in the scale of observation or application, has examples in everyday life which provide useful metaphors for this facet of the quantummechanical model. 19 One methodology suggested by the ontological characterization of the quantum situation bases itself on the axiomatic formulations of modern mathematics. Given that (1) the descriptive images lose their representative significance in the microphysical domain and (2) the unity or consistency possessed by the quantum model is of a strictly mathematical kind, the impulse is then to regard and to develop quantum theory as an axiomatic system, in the mathematical sense, and proceed to test the deduced theorems against empirical evidence via formal rules of correspondence.

P . 220. 19 B R A I T H W A I T E , R. B. Scientific Explanation, Cambridge University Press, 1953, p . 93. 20 BLALOCK, H. M . Causal Inferences in Non-experimental Research, Chapel Hill, University of North Carolina Press, 1965, p . 6. 21 T h e issue is explicitly set out already HIDESCARTES, R. Meditations on the First Philosophy written at the beginning of the sixteenth century. 22 A P O S T E L , L. 'Towards the Formal Study of Models in non-formal Sciences', in F R E D E N T H A L , H . The Concept and the Role of Model in Mathematics and Natural and Social Sciences, Dordtrecht, 1961, p .