Local and Global Aspects of Lie's Superposition Theorem

Details Local and Global Aspects of Lie's Superposition Theorem Local and Global Aspects of Lie's Superposition Theorem

2009-01-28

arxiv.org/abs/0901.4478v2

Mathematics

Classical Analysis and ODEs

43 pages

Scientific paper

In this paper we give the global conditions for an ordinary differential equation to admit a superposition law of solutions in the classical sense. This completes the well-known Lie superposition theorem. We introduce rigorous notions of pretransitive Lie group action and Lie-Vessiot systems. We proof that pretransitive Lie group actions are transitive. We proof that an ordinary differential equation admit a superposition law if and only if it is a pretransitive Lie-Vessiot system. It means that its enveloping algebra is spanned by fundamental fields of a pretransitive Lie group action. We discuss the relationship of superposition laws with differential Galois theory and review the classical result of Lie.

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