On Lie systems and Kummer-Schwarz equations

Abstract

A Lie system is a system of first-order differential equations admitting a superposition rule, i.e., a map that expresses its general solution in terms of a generic family of particular solutions and certain constants. In this work, we use the geometric theory of Lie systems to prove that the explicit integration of second- and third-order Kummer--Schwarz equations is equivalent to obtaining a particular solution of a Lie system on SL(2,R). This same result can be extended to Riccati, Milne--Pinney and other related equations. We demonstrate that all the above-mentioned equations associated with exactly the same Lie system on SL(2,R) can be integrated simultaneously. This retrieves and generalizes in a unified and simpler manner previous results appearing in the literature. As a byproduct, we recover various properties of the Schwarzian derivative.

29 pages. A relevant error and several typos corrected

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Cited by in corpus (8)

• Lie-Hamilton systems on the plane: properties, classification and applications
• Dirac--Lie systems and Schwarzian equations
• k-symplectic Lie systems: theory and applications
• Poisson-Hopf algebra deformations of Lie-Hamilton systems
• Contact Lie systems
• A geometric Hamilton--Jacobi theory for a Nambu--Poisson structure
• Multisymplectic structures and invariant tensors for Lie systems
• Lie-Hamilton systems on Riemannian and Lorentzian spaces from conformal transformations and some of their applications