2009/01/28 by David Blázquez-Sanz, Blázquez-Sanz, David, Juan José Morales-Ruiz +1
Mathematics · Physics and Astronomy · #34A26 #Advanced Topics in Algebra #Classical Analysis and ODEs (math.CA) #Dynamical Systems (math.DS) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Nonlinear Waves and Solitons #math.CA #math.DS #msc:34A26
paper · pdf · doi:10.48550/arxiv.0901.4478
43 pages
openalex publication_date 2009/01/28 · arxiv created 2009/11/06 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/03
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.