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.
- Publication:
-
Journal of Mathematical Physics
- Pub Date:
- March 2013
- DOI:
- 10.1063/1.4794280
- arXiv:
- arXiv:1212.5779
- Bibcode:
- 2013JMP....54c3505D
- Keywords:
-
- 02.30.-f;
- 02.40.-k;
- 02.60.Jh;
- 02.20.Sv;
- Function theory analysis;
- Geometry differential geometry and topology;
- Numerical differentiation and integration;
- Lie algebras of Lie groups;
- Mathematical Physics;
- Mathematics - Classical Analysis and ODEs;
- 34A26
- E-Print:
- 29 pages. A relevant error and several typos corrected