The Journal of Geometric Mechanics (JGM)

Superposition rules and second-order Riccati equations

Pages: 1 - 22, Volume 3, Issue 1, March 2011      doi:10.3934/jgm.2011.3.1

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José F. Cariñena - Departamento de Física Teórica and IUMA, Facultad de Ciencias, Universidad de Zaragoza, Pedro Cerbuna 12, 50.009, Zaragoza, Spain (email)
Javier de Lucas Araujo - Institute of Mathematics, Polish Academy of Sciences, Śniadeckish 8, P.O. Box 21, 00-956, Warszawa, Poland (email)

Abstract: A superposition rule is a particular type of map that enables one to express the general solution of certain systems of first-order ordinary differential equations, the so-called Lie systems, out of generic families of particular solutions and a set of constants. The first aim of this work is to propose various generalisations of this notion to second-order differential equations. Next, several results on the existence of such generalisations are given and relations with the theories of Lie systems and quasi-Lie schemes are found. Finally, our methods are used to study second-order Riccati equations and other second-order differential equations of mathematical and physical interest.

Keywords:  Quasi-Lie scheme, Lie system, system of second-order differential equations, second-order Riccati equation, superposition rule.
Mathematics Subject Classification:  34A26, 34A34, 53Z05.

Received: June 2010;      Revised: April 2011;      Available Online: April 2011.