Local and Global Aspects of Lie's Superposition Theorem

Abstract

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.

43 pages

References in corpus (2)

• Superposition rules, Lie theorem and partial differential equations
• Differential Galois Theory of Algebraic Lie-Vessiot Systems

Cited by in corpus (2)

• On Lie systems and Kummer-Schwarz equations
• A Quasi-Lie Schemes Approach to Second-Order Gambier Equations